GIFT  OF 
Dr.   Horace   Ivie 


EDUCATION  DEPT 


HAY'S    MATHEMATICAL    SERIES. 


SURVEYING 


NAVIGATION,  :  :  v  j  :  \ ;       ;';, 


WITH   A  PRELIMINARY  TREATISE  ON 


TRIGONOMETRY  AND  MENSURATION, 


A.  §CHUYLER,  M.  A. 


Professor  of  Applied  Mathematics  and  Logic  in  Baldwin  University;  Author  of 
Higher  Arithmetic,  Principles  of  Loyic,  and  Complete  Algebra. 


VAN  ANTWERP,  BRAGG  &  CO., 
137  WALNUT  STREET,  28  BOND  STREET, 

CINCINNATI.  NEW  YORK. 


RAY'S   SERIES, 


EMBRACING 


A  Thorough  and  Progressive  Course  in  Arithmetic,  Algebra, 
the  Higher  Mathematics. 


Primary  Arithmetic.  Higher  Arithmetic. 

/^InteHect^i&L Arithmetic.  Test  Examples  in  Arithmetic. 

Rntiiiiients  01  Arithmetic.  New  Elementary  Algebra. 

Practical  AHtiwiietic.  New  Higher  Algebra. 


Plane  and  Noliri  Geometry.    BY   ELI  T.  TAPPAN,  A.M.,  Preset 

Kenyan  College.     12wo,  cloth,  276  pp. 
Geometry    and    Trigonometry     By   ELI    T.   TAPPAN,   A.M. 

I- res' t  Kenyan   College.    Svo,  sheep,  420  pp. 
Analytic  Geometry.     By  GEO.  H.  HOWISON,  A.M.,  Prof,  in  Mass. 

Institute  of  Technology.    Treatise  on  Analytic  Geometry,  especially 

as  applied  to  the   Properties  of   Conies :    including  the  Modern 

Methods  of  Ahridged  Notation. 
Elements  of  Astronomy.    By  S.  H*"PEABODY,  A.M.,  Prof,  of 

Physics  and  Civil  Engineering,  Amherst  College.      Handsomely  and 

profusely  illustrated.     8vo,  sheep,  336  pp. 


KEYS. 

Ray's  Arithmetical  Key  (To  Intellectual  and  Practical); 
Key  to  Ray's  Higher  Arithmetic ; 

Key  to  Ray's  New  Elementary  and  Higher  Algebras, 
GIFT  OF 


The  Publishers  furnish  Descriptive  Circulars  of  the  above  Mathe- 
matical Text-BooJtSf  with  Prices  and  other  information  concerning 
them. 

_ 


Entered  according  to  Act  of  Congress,  in  the  year  1864,  by  SARGENT,  WILSON  & 

HINKLE,  in  the  Clerk's  Office  of  the  District  Court  of  the  United 

States  for  the  Southern  District  of  Ohio. 


PREFACE. 


Nearly  twenty  years  ago  the  Publishers  made  the  following 
announcement:  "Surveying  and  Navigation;  containing  Survey- 
ing and  Leveling,  Navigation,  Barometric  Heights,  etc." 

To  redeem  this  promise,  the  present  work  now  appears. 

It  is  customary  to  preface  works  on  Surveying  by  a  meager 
sketch  of  Plane  Trigonometry,  but  it  has  been  thought  best 
to  include  in  this  work  a  thorough  treatment  of  Plane  and 
Spherical  Trigonometry  and  Mensuration.  These  subjects  have 
been  treated  in  .view  of  the  wants  of  our  best  High  Schools 
and  Colleges. 

Certain  modern  writers  have  defined  the  Trigonometric  func- 
tions as  ratios;  for  example,  in  a  right  triangle,  the  sine  of  an 
angle  is  the  ratio  of  the  opposite  side  to  the  hypotenuse,  etc. 

The  historical  method  of  considering  the  sine,  co-sine,  tan- 
gent, etc.,  as  linear  functions  of  the  arc,  explains  the  origin  of 
these  terms  —  avoids  the  ambiguity  of  the  word  ratio;  explains 
how  the  logarithm  of  the  sine,  for  example,  can  reach  the  limit 
10,  which  would  be  impossible  if  the  limit  of  the  sine  itself  is 
1,  and  is  much  more  readily  apprehended  by  the  student. 

The  advantages  in  analytic  investigations  resulting  from 
defining  these  functions  as  ratios  have  been  secured  in  the 
principles  relating  to  the  Eight  Triangle,  Art.  64. 

Each  of  the  circular  functions  has,  in  the  first  place,  been 
considered  by  itself,  and  its  value  traced,  for  all  arcs,  from  0° 
to  360°. 

924229         <««) 


iv  PREFACE. 

Then  follows  the  solution  of  triangles,  right  and  oblique,  the 
general  relations  of  the  circular  functions,  the  functions  of  the 
sum  or  difference  of  two  angles,  and  a  variety  of  interesting 
practical  applications. 

It  is  hoped  that  Spherical  Trigonometry  has  been  made  in- 
telligible to  the  diligent  student.  More  than  ordinary  care  has 
been  given  to  the  development  of  Napier's  principles,  and  to 
the  discussion  of  the  species  of  the  parts  of  both  right  and 
oblique  spherical  triangles,  Arts.  126,  129,  145,  148,  151. 

Mensuration,  a  subject  at  once  interesting  and  practically  im- 
portant, has  been  discussed  at  length,  and  formulas  have  been 
developed  instead  of  rules  for  the  solution  of  the  problems. 

In  the  Surveying,  the  instruments  are  first  represented  and 
described,  and  the  methods  of  making  the  adjustments  given 
in  detail. 

The  Author  takes  this  opportunity  to  express  his  obligations 
to  Messrs.  W.  &  L.  E.  Gurley,  Manufacturers  of  Surveying  and 
Engineering  Instruments,  Troy,  N.  Y.,  who  have  kindly  granted 
him  the  use  of  their  Manual  for  the  delineation  and  descrip- 
tion of  the  instruments.  In  consequence  of  this  courtesy,  much 
better  drawings  and  descriptions  have  been  made  than  would 
otherwise  have  been  possible. 

The  instruments  themselves  should,  however,  be  accessible  to 
the  student,  who  should  study  them  in  connection  with  the 
descriptions  in  the  book,  and  learn  to  use  them  in  practical 
work,  guided  by  a  competent  instructor. 

The  Rectangular  method  of  surveying  the  Public  lands,  now 
brought  to  great  perfection  under  the  direction  of  the  Govern- 
ment, has  been  minutely  explained,  and  illustrated  by  field 
notes  of  actual  surveys.  In  this  portion  of  the  work,  the 
United  States  Manual  of  Surveying  Instructions  has  been 
taken  as  authority,  and  thus  the  authorized  methods,  which 
must  form  the  basis  for  subsequent  surveys,  have  been  made 
accessible  to  the  student. 

The  methods  of  finding  the  true  meridian  and  the  variation 
of  the  needle  have  been  given  at  length;  also  specific  direc- 


PREFACE.  V 

tions  for  finding  corners,  taking  bearings,  measuring  lines,  re- 
cording field  notes,  and  plotting. 

In  addition  to  the  ordinary  method  of  finding  the  area,  a 
new  method,  developed  by  E.  M.  Pogue,  of  Kentucky,  is  given 
in  Art.  304.  This  method  has  the  merit  of  giving  always  a 
uniform  result  from  the  same  field  notes,  and  thus  avoids  dis- 
putes about  the  different  results  of  the  ordinary  method,  un- 
avoidably attending  the  various  distribution  of  errors  by  differ- 
ent calculators. 

The  methods  of  supplying  omissions  are  explained  and  illus- 
trated by  examples. 

Laying  out  and  dividing  land,  operations  admitting  of  an 
unlimited  variety  of  applications,  have  been  treated  in  view  of 
the  wants  of  the  practical  surveyor.  The  subject  is  also  full  of 
interest  to  the  student,  who  can  not  fail  to  receive  from  it  new 
views  of  the  resources  of  mathematical  science. 

Leveling,  the  construction  of  railroad  curves,  embankments 
and  excavations,  the  method  of  making  Topographical  surveys, 
with  the  authorised  conventional  symbols,  Barometric  heights, 
etc.,  have  been  explained  and  illustrated  by  diagrams  and 
examples. 

It  has  been  thought  best  to  give  a  clear,  elementary  treat- 
ment of  Navigation,  not  only  on  account  of  those  who  may 
desire  to  pursue  the  subject  further,  but  for  the  sake  of  grati- 
fying the  wishes  of  intelligent  persons  who  may  desire  to  know 
something  of  Navigation.  The  limits  of  the  work,  however,  for- 
bid the  discussion  of  Nautical  Astronomy.  The  examples  in 
Navigation  have  been  selected  from  the  English  work  of  J.  R. 
Young. 

The  tables  of  Logarithms,  Natural  and  Logarithmic  sines,  etc., 
have  been  carried  only  to  five  decimal  places,  and  for  the  pur- 
poses intended  will  be  found  practically  better  than  tables  to 
six  or  seven  places. 

The  Traverse  table  has  been  thrown  into  a  new  form,  at  once 
condensed  and  convenient. 

These  tables  have  been  compiled  by  Mr.  Henry  H.  Vail,  and 


VI  PREFACE. 

by  him  compared  with  Babbage's  and  Wittstem's  tables,  then 
by  the  Author  with  Vega's  tables  to  seven  decimal  places.  It 
is  hoped  that  by  this  double  comparison  perfect  accuracy  has 
been  attained. 

The  table  of  Meridional  Parts,  taken  from  "  Projection  Tables  for 
the  use  of  the  United  States  Navy,"  prepared  by  the  Bureau  of 
Navigation,  and  issued  from  the  Government  Printing  office,  was 
calculated  in  the  Hydrographic  office  for  the  terrestrial  spheroid, 
compression  u^.ir?¥-  This  table,  now  for  the  first  time  pub- 
lished in  a  text-book,  is  believed  to  be  more  correct  than  those  in 
general  use. 

The  Author  takes  pleasure  in  acknowledging  his  obligations  to 
Prof.  E.  H.  Warner  for  critical  suggestions  and  acceptable  aid  in 
reading  proof  and  testing  the  accuracy  of  the  answers. 

With  the  hope  that  the  book  will  be  attractive  and  useful  to 
the  student,  teacher,  and  practical  surveyor,  it  is  sent  forth  to 
accomplish  its  work. 

A.  SCHUYLER. 
BALDWIN   UNIVERSITY.     ) 
BEREA,  0.,  June  12,  1873.  j 


INDEX. 


PAGE 

INTRODUCTION 9 

Logarithms 9 

Table  of  Logarithms .         .  12 

Multiplication  by  Logarithms      .......  18 

Division  by  Logarithms 19 

Involution  by  Logarithms     .         .         .         .         .         .         .         .21 

Evolution  by  Logarithms      ........  22 

TRIGONOMETRY 23 

Plane  Trigonometry .23 

Trigonometrical  Functions .27 

Table  of  Natural  Functions 41 

Table  of  Logarithmic  Functions 43 

Right  Triangles 47 

Oblique  Triangles '.        .      55 

Application  to  Heights  and  Distances          .         ...         .         .69 

Relations  of  Circular  Functions  .......       72 

Applications 92 

SPHERICAL,  TRIGONOMETRY      . 108 

Right  Triangles 109 

Oblique  Triangles          .        . 124 

Mensuration .         .         .         .150 

Mensuration  of  ^Surfaces  • 150 

Mensuration  of  Volumes      ........     174 

SURVEYING 185 

Instruments 185 

Survey  of  Public  Lands 216 

Variation  of  the  Needle .265 

Field  Operations 274 

Preliminary  Calculations .        .  284 

Area  of  Land 296 

(vii) 


viii  INDEX. 

PAGE 

Supplying  Omissions 308 

Laying  Out  Land .  313 

Dividing  Land 318 

Leveling 340 

Surveying  Railroads      .........  351 

Topographical  Surveying      .         .         .         .         .         .         .         .  369 

Barometric  Heights 375 

NAVIGATION 381 

Preliminaries 381 

Plane  Sailing 385 

Parallel  Sailing 389 

Middle  Latitude  Sailing 390 

Mercator's  Sailing          .                 393 

Current  Sailing 398 

Plying  to  Windward .        .        .  400 

Taking  Departures 402 

TABLES                                                                                      ,  405 


INTRODUCTION  i  V,     Oi: 


LOGARITHMS. 

1.    Definition. 

A  logarithm  of  a  number  is  the  exponent  denoting 
the  power  to  which  a  fixed  number,  called  the  base,  must 
be  raised  in  order  to  produce  the  given  number. 

Thus,  in  the  equation,  bx  =  n,  b  is  the  base  of  the  sys- 
tem, n  is  the  number  whose  logarithm  is  to  be  taken, 
and  x  is  the  logarithm  of  n  to  the  base  6,  which  may  be 
written  :  x  =  log  b  n. 

Any  positive  number,  except  1,  may  be  assumed  as 
the  base,  but  when  assumed,  it  remains  fixed  for  a  sys- 
tem ;  hence,  there  may  be  an  infinite  number  of  sys- 
tems, since  there  may  be  an  infinite  number  of  bases. 

2.   Common  Logarithms. 

Common  logarithms  are  the  logarithms  of  numbers 
in  the  system  whose  base  is  10. 

10°  =  lj.  .'.  by  def.,  log  1=0.' 

101  =10;  .;.  by  def.,  log  10=1. 

102  =  100 ;  .  • .  by  def.,  log  100  =  2. 

103  =  1000;  . ' .  by  def.,  log  1000  =  3. 


Hence,  In  the  common  system,  the  logarithm  of  an  exact  power 
of  10  is  the  whole  number  equal  to  the  exponent  of  the  power. 

(9) 


10  LOGARITHMS. 

3.   Consequences. 

1.  If  the  number  is  greater  than  1  and  less  than  10, 
its  logarithm  is  greater  than  0  and  less  than  1,  or  is  0 
a  decimal.  • 

2f'If  thV  nuuiber  is  greater  than  10  and  less  than 
*  '10$,  its'  logarithm  is  greater  than  1  and  less  than  2,  or 
is  1  -}-  a  decimal. 

3.  In  general,  if  the  number  is  not  an  exact  power 
of  10,  its  logarithm,  in  the  common  system,  will  consist 
of  two  parts — an  entire  part  and  a  decimal  part. 

The  entire  part  is  called  the  characteristic  and  the  dec- 
imal part  is  called  the  mantissa. 

4:.    Problem. 

To  find  the  laws  for  the  characteristic. 

.    Let  (1)  10Z  =  n  •  then,  by  def.,  log  n  =  x. 
But  (2)  lO^lO. 

(l)-(2)  =  (3)  10*-'  =  ^;  then,  by  def,  log  ^  =  x-l. 
•  *  •     log  ^  =  log  n  —  1. 

Hence,  The  logarithm  of  the  quotient  of  any  number  by  10  -is 
less  by  1  than  the  logarithm  of  the  number. 

Let  us  now  take  the  number  8979  and  its  logarithm 
3.95323,  as  given  in  a  table  of  logarithms,  and  divide 
the  number  successively  by  10,  and  Tor  each  division 
subtract  1  from  the  logarithm  of  the  dividend,  then  we 
have, 

Log  8979  ^3.95323.  Log  .8979       =T95323. 

"  897.9  ~  2.95323.  "  .08979    =^95323. 

"  89.79  =  L95323.  u  .008979  —  3795323. 

"  8.979  =  0.95323. 


THE  CHARACTERISTIC.  11 

The  minus  sign  applies  only  to  the  characteristic  over 
which  it  is  placed. 

The  mantissa  is  always  positive,  and  is  the  same  for 
all  positions  of  the  decimal  point. 

An  inspection  of  the  above  will  reveal  the  following 
laws  : 

1.  If  the  number  is  integral  or  mixed,  the  characteristic  is 
positive  and  is  one  less  than  the  number  of  integral  figures. 

2.  If  the  number  is  entirely  decimal,  the  characteristic  is 
negative  and  is  one  greater,  numerically,  than  the  number  of 
O's  immediately  following  the  decimal  point. 

5.    Exercises  on  the  Characteristic. 

1.  What  is  the  characteristic  of  the  logarithm  of  7? 

2.  What  is  the   characteristic   of  the  logarithm   of 
465? 

3.  What  is  the   characteristic   of  the  logarithm   of 
4678? 

4.  What  is   the   characteristic   of  the  logarithm   of 
34.75? 

5.  What  is   the   characteristic   of  the  logarithm   of 
.65? 

6.  What  is   the   characteristic   of  the  logarithm  of 
.0789? 

7.  What  is   the   characteristic   of  the  logarithm   of 
.00084? 

8.  If  the  characteristic  of  the  logarithm  of  a  num- 
ber is  2,  how  many  integral  places  has  that  number? 

9.  If  the  characteristic  of  the  logarithm  of  a  num- 
ber is  5,  how  many  integral  places  has  that  number? 

10.  If  the  characteristic  of  the  logarithm  of  a  num- 
ber is  1,  how  many  integral  places  has  -that  number? 

11.  If  the  characteristic  of  the  logarithm  of  a  num- 
ber is  0,  how  many  integral  places  has  that  number? 


12  LOGARITHMS. 

12.  If  the  characteristic  of  the  logarithm  of  a  num- 
ber  is   negative,   is   the    number    integral,   decimal,   or 
mixed? 

13.  If  the  characteristic  of  the  logarithm  of  a  num- 
ber is  4,  how  many  O's  immediately  follow  the  decimal 
point? 

14.  If  the  characteristic  of  the  logarithm  of  a  num- 
ber is  2,  how  many  O's  immediately  follow  the  decimal 
point? 

15.  If  the  characteristic  of  the  logarithm  of  a  num- 
ber is  1,  how  many  O's  immediately  follow  the  decimal 
point  ? 

TABLE  OF  LOGARITHMS. 

6.  Description  of  the  Table. 

The  table  of  logarithms  annexed  gives  the  mantissa 
of  the  logarithm  of  every  number  from  1000  to  10900. 
The  characteristic  can  be  found  by  the  preceding  laws. 

It  follows,  from  Art.  4,  that  the  mantissa  of  the  loga- 
rithm of  a  number  is  the  same  as  the  mantissa  of  the 
logarithm  of  the  product  or  quotient  of  that  number  by 
any  power  of  10.  Thus : 

Log  12  :  =  1.07918. 
"  120  =  2.07918. 
"  .012  =  2X)7918: 

Hence,  we  can  determine  from  the  table  the  log- 
arithm of  aii£  number  less  than  1000.  Thus,  the 
mantissa  of  the  logarithm  of  8  is  the  same  as  that 
of  the  logarithm  of  8000. 

In  the  table,  the  first  three  or  four  figures  of  each 
number  are  given  in  the  left-hand  column,  marked 
X.  The  next  figure  is  given  at  the  head  and  foot 
of  one  of  the  columns  of  mantissas. 


TABLE  OF  LOGARITHMS.  13 

The  mantissas,  in  the  column  under  0,  are  given 
to  five  decimal  places.  The  first  and  second  decimal 
figures  of  this  column  are  understood  to  be  repeated  in 
the  spaces  below,  and  to  be  prefixed,  across  the  page, 
to  the  three  figures  of  the  remaining  columns. 

When  the  third  decimal  digit  changes  from  9  to  0, 
the  second  is  increased  by  the  1  carried;  and  the  cor- 
responding mantissa,  and  all  to  the  right,  commence 
with  a  smaller  figure,  to  indicate  that  the  first  two 
decimal  figures,  to  be  prefixed,  are  to  be  taken  from 
the  line  below. 

The  last  column,  marked  D,  contains  the  differ- 
ence of  two  successive  mantissas,  called  the  tabular 
difference. 

7.    Problem. 
To  find  the  logarithm  of  a  given  number. 

1.  Find  the  logarithm  of  3675. 

The  characteristic  is  3.  Opposite  367,  in  the  column 
headed  N,  and  under  the  column  headed  5,  we  find 
526,  to  which  prefix  the  two  figures,  56,  in  the  column 
headed  0,  and  we  have  for  the. mantissa  .56526. 

.'.   log  3675  r=  3.56526. 

2.  Find  the  logarithm  of  76. 

The  characteristic  is  1,  and  the  mantissa  is  the  same 
as  that  of  7600,  which  is  .88081. 

.  • .    log  76  =  1.88081. 

3.  Find  the  logarithm  of  .004268. 

The  characteristic  is  3,  and  the  mantissa  is  the  same 
as  that  of  4268.  Looking  opposite  426,  and  under  8, 
we  find  022,  of  which  the  0  is  a  small  figure.  Prefixing 


14  LOGARITHMS. 

63,  from  the  line  below,  in  the  column  headed  0,  we 
have  for  the  mantissa  .63022. 

.-.     log   .004268  ="3.63022. 

4.  Find  the  logarithm  of  109684. 

The  characteristic  =5. 

The  mantissa  of  log    1096      =    .03981 
Tab.  diff.  is  40;  and  40  X  .84  «a          34 

log   109684  =  5.04015 

The  reason  for  multiplying  the  tabular  difference  by 
.84  will  be  apparent  from  the  following: 

log    109600  =  5.03981. 
log   109700  =  5.04021. 

The  difference  of  the  logarithms  is  40  hundred- 
thousandths,  and  the  difference  of  the  numbers  is  100; 
but  the  difference  of  109600  and  109684  is  84,  which  is 
.84  of  100;  hence,  the  difference  of  the  logarithms  of 
109600  and  109684  is  .84  of  40  hundred-thousandths, 
which  is  40  hundred-thousandths  X  -84  =  34  hundred- 
thousandths,  nearly. 

It  is  assumed  that  the  difference  of  the  logarithms 
of  two  numbers  is  proportional  to  the  difference  of  the 
numbers,  which  is  approximately  true,  especially  if  the 
numbers  are  large. 

5.  Find  the  logarithm  of  123.613. 

The  characteristic  =  2. 

The  mantissa  of  log    1236     =    .09202 
Tab.  diff.  is  35;   and  3oX-13=  5 

.;.,     log   123.613  =  2.09207 

The  tabular  difference  is  .00035,  and  .00035  X  .13  = 
.0000455.  But  since  the  logarithms  in  this  table  are 
taken  only  to  five  decimal  places,  the  two  last  figures, 


EXAMPLES.  15 

55,  are  rejected,  and  1  is  carried  to  .00004,  making 
.00005  for  the  correction. 

In  general,  when  the  left-hand  figure  of  the  part 
rejected  exceeds  4,  carry  1. 

When  the  tabular  difference  is  large,  as  in  the  first 
part  of  the  table,  there  may  be  small  errors.  Accord- 
ingly, for  numbers  between  10000  and  10900,  it  will 
be  better  to  use  the  last  two  pages  instead  of  the 
first  page. 

8.    Rule. 

1.  If   the  number,  or  the  product  of  the  number   by  any 
power  of  W,  is  found  in  the  table,  take  the  corresponding  man- 
tissa from  the  table,  and  prefix  the  proper  characteristic. 

2.  If  the  number,  without  reference  to   the.  decimal  point 
or  O'.s  on  the  right,  is   expressed   by  more   than  five  figures, 
take  from   the  table  the  mantissa   corresponding  to  the  first 
four  or  five  figures  on   the  left,  multiply  the  corresponding 
tabular  difference  by  the  number  expressed  by  the  remaining 
figures,  considered  as  a  decimal,  reject  from  the  product  as 
many  figures  on  the  right  as  are  in  the  multiplier,  carrying 
to  the  nearest  unit,  and  add  the  result  as  so  many  hundred- 
thousandths  to   the  mantissa   before  found,  and    to    the  sum 
prefix  the  proper  characteristic. 

9.    Examples. 

1.  What  is  the  logarithm  of  2347  ?  Ans.  3.37051. 

2.  What  is  the  logarithm  of  108457?  Ans.  5.03526. 

3.  What  is  the  logarithm  of  376542?  Ans.  5.57581. 

4.  What  is  the  logarithm  of  229.7052?  Ana.  2.36117. 

5.  What  is  the  logarithm  of  1128737?  Ans.  6.05260. 

6.  What  is  the  logarithm  of  .30365?  Ans.  L48237. 

7.  What  is  the  logarithm  of  .0042683?  Ana.  "3.63025. 

8.  What  is  the  logarithm  of  1245400?  Ans.  6.09531. 


16  LOGARITHMS. 

10.   Problem. 

To  find  the  number  corresponding  to  a  given  logarithm. 

1.  What  number  corresponds  to  logarithm  2.03262? 
The    mantissa    is    found    in    the    column    headed    8, 

and  opposite  107  in  the  column  headed  N.  Hence, 
without  reference  to  the  decimal  point,  the  number 
corresponding  is  1078;  but  since  the  characteristic  is 
2,  the  number  is  entirely  decimal,  and  one  0  imme- 
diately follows  the  decimal  point.  Hence,  the  number 
corresponding  is  .01078. 

2.  What  number  corresponds  to  logarithm  2.83037? 
Since  this  logarithm  can  not  be  found  in  the  table, 

take  the  next  less,  which  is  2.83033,  and  the  corre- 
sponding number,  without  reference  to  the  decimal 
point,  which  is  6766. 

The  difference  between  the  given  logarithm  and  the 
next  less  is  4,  and  the  tabular  difference  is  6,  which 
is  the  difference  of  the  logarithms  of  the  two  numbers, 
6766  and  6767,  whose  difference  is  1. 

If  the  tabular .  difference .  of  the  logarithms,  6,  cor- 
responds to  a  difference  in  the  numbers  of  1,  the 
difference  of  the  logarithms,  4,  will  correspond  to  a 
difference  of  -f  of  1 ;  which,  reduced  to  a  decimal,  and 
annexed  to  6766,  will  give  for  the  number,  without 
reference  to  the  decimal  point,  676666.  But  since  the 
characteristic  is  2,  there  will  be  three  integral  places; 
hence,  676.666  is  the  number  required. 

3.  What  number  corresponds  to  logarithm  2.76398? 

The  given  log  =2.76398      .  •.  number  =  580.737 
Next  less  log    =  2.76395      .  • .  number  =  580.7 
Tab.  difference  ct=  8)300  =  difference. 
37  =  correction. 


TABLE  OF  LOGARITHMS.  17 

It  is  necessary  to  write  only  that  part  of  the  next 
less  logarithm  which  differs  from  the  given  logarithm. 
Conceive  O's  annexed  to  the  difference,  and  divide  by 
the  tabular  difference;  and  annex  the  quotient  to  the 
number  corresponding  to  the  next  less  logarithm. 

In  practical  work  abbreviate  thus :  Let  I  denote  the 
given  logarithm ;  /',  the  next  less  logarithm ;  n  and  n', 
the  corresponding  numbers;  t,  the  tabular  difference; 
dj  difference  of  logarithms;  c,  the  correction. 

4.  What  number  corresponds  to  logarithm  1.73048? 

I  =1.73048    r ;.  n  ==  .537625 
P=T.73Q46    .'.  n'=  .5376 

t  =  8)  2  —  dL  n'  is  found  first,  then 

25  =  c.  n  by  annexing  c. 

11.    Rule. 

1.  If  the  given  mantissa  can  be  found  in  the  table,  take  the 
number  corresponding,  and  place  the  decimal  point  accord- 
ing to  the  law  for  the  characteristic. 

2.  If  the  given   mantissa  can   not  be  found  in  the  table, 
take  the  next  less  and  the  corresponding  number.     Subtract 
this  mantissa   from  the  given  mantissa,  annex  O's  to  the  re- 
mainder, divide  the  result  by  the  tabular  difference,  annex 
the  quotient   to  the  number  corresponding  to  the  logarithm 
next  less  than   the  given   logarithm,  and  place  the  decimal 
point  according  to  the  law  for  the  characteristic. 

12.   Examples. 

1.  What  number  corresponds  to  logarithm  ,4.55703? 

Ans.  36060. 

2.  What  number  corresponds  to  logarithm  3.95147? 

Ans.  8942.8. 

3.  What  number  corresponds  to  logarithm  2.41130? 

Ans.  .025781. 
S.  N.  2. 


18  LOGARITHMS. 

4.  What  number  corresponds  to  logarithm  1.48237? 

Ans.  .30365. 

5.  What  number  corresponds  to  logarithm  3.63025? 

Ans.  .0042683. 


MULTIPLICATION  BY  LOGARITHMS. 
13.    Proposition. 

The  logarithm  of  the  product  of  two  numbers  is  equal  to 
the  sum  of  their  logarithms. 

(1)  bx  =  m;  then,  by  def.,  log  m  .=  x. 
Let 

(2)  by  =  n;  then,  by  def.,  log  n  =  y. 


(1)X(2)  =  (3)   b*  +  y  =  mn;  then,  by  def.,  log  mn=x+y. 

.  '  .     log  m  n  =  log  m  -f-  log  n. 

14.   Rule. 

1.  Find  the  logarithms  of  the  factors  and  take  their  sum, 
which  will  be  the  logarithm  of  the  product. 

2.  Find  the   number   corresponding   which  will   be   their 
product. 

15.    Examples. 

1.  Find  the  product  of  57846  and  .003927. 

log  57846  =  4.76228 
log  .003927  ="3.59406 
log  product  =  2.35634,  .  •  .  product  =  227.16. 

2.  Find  the  product  of  37.58  and  75864. 

Ans.  2851000. 

3.  Find  the  product  of  .3754  and  .00756. 

Ans.  .002838. 


DIVISION  BY  LOGARITHMS.  19 

4.  Find  the  product  of  999.75  and  75.85. 

Ans.  75831.667. 

5.  Find  the  product  of  85,  .097,  and  .125.    Ans.  1.03062. 

DIVISION   BY   LOGARITHMS. 
16.   Proposition. 

The  logarithm  of  the  quotient  of  two  numbers  is  equal  to 
the  logarithm  of  the  dividend  minus  the  logarithm  of  tlie 
divisor. 

C  (1)     b*=  m;  then,  by  def.,  log  m  =  x. 
Let   ] 

(  (2)     6 y  --  n;  then,  by  def,  log  n  =  y. 

(1) -5- (2)  =;  (3)     &*-»=—;    then,  by  def.,  log-  =  x  —  y. 

i      m       i 
.  • .     log  —  —  log  m  —  log  n. 

17.   Rule. 

1.  Find  the  logarithms  of  the  numbers,  subtract  the  loga- 
rithm of  the  divisor  from  the  logarithm  of  the  dividend,  and 
the  remainder  will  be  the  logarithm  of  the  quotient. 

2.  Find    the    number    corresponding    which    ivill    be    the 
quotient. 

18.   Examples. 

1.  Divide  73.125  by  .125. 

log  73.125=1.86407 
log  .125  =  L09691 
log  quotient  =  2.76716,  .  • .  quotient  =  585. 

2.  Divide  7.5  by  .000025.  Ans.  300000. 

3.  Divide  87.9  by  .0345.  Ans.  2547.824. 

4.  Divide  .34852  by  .00789.  Ans.  44.171. 

5.  Divide  85734  bv  12.7523.  Ans.  6723. 


20  LOGARITHMS. 

ARITHMETICAL  COMPLEMENT.- 

19.    Definition. 

The  arithmetical  complement  of  a  logarithm   is  the 
result  obtained  by  subtracting  that  logarithm  from  10. 
Thus,  denoting  the  logarithm  by  /.,  and  its  arithmetical 
complement  by  a.  c.  /.,  we  shall  have  the  formula, 
a.  c.  I  =10^-1. 

The  arithmetical  complement  of  a  logarithm  is  most 
readily  found  by  commencing  at  the  left  of  the  loga- 
rithm, and  subtracting  each  digit  from  9  till  we  come 
to  the  last  numeral  digit,  which  must  be  subtracted 
from  10. 

Thus,  to  find  the  a.  c.  of  3.47540,  we  say:  3  from  9, 
6;  4  from  9,  5;  7  from  9,  2;  5  from  9,  4;  4  from  10,  6; 
0  from  0,  0. 

.-.    a.  c.  of  3.47540  =  6.52460. 

20.  Proposition. 

The  difference  of  two  logarithms  is  equal  to  the  minuend, 
plus  the  arithmetical  complement  of  the  subtrahend,  minus  10. 

For,  I  —  Z'=Z  +  (10  —  I'}  — 10. 

It  is  convenient  to  use  the  a.  c.  in  division  when 
either  the  dividend  or  the  divisor  is  the  indicated 
product  of  two  or  more  factors.  Thus,  let  it  be  re- 
quired to  find  x  in  the  proportion : 

37.5:  678.5::  27.56:,;    ...,  =  678.5  x27.56 

o7.o 
,  • ,     log  x  =  log  678.5  -f  log  27.56  -f  a.  c.  log  37.5  —  10. 

log  678.5  =  2.83155 

log  27.56=1.44028 

a.  c.  log    37.5  =  8.42597 

log    x      =  2.69780     .  • .  x  =  498.656. 


INVOLUTION  BY  LOGARITHMS.  21 

21.    Examples. 

1.  Given  125.5  :  .0756  : :  x  :  .0034532,  to  find  x. 

Ans.  5.7325. 

2.  Given  843  :  x  : :  732.534  :  .759,  to  find  x. 

Ans.  .87346. 

3.  Given  x  :  .034  : :  .784  :  .00489,  to  find  x, 

Ans.  5.451125. 
32.015  X-.874 
4  °1Ven  X  **  .000216X90257 '  t0  find  *      An°'  L4358' 

5.  Given  .753  X  12.234  :  87.5  X  3.7547  : :  56.5  :  x,  to 
find  x.  Ans.  2014.96. 

INVOLUTION   BY   LOGARITHMS. 
22.    Proposition. 

The  logarithm  of  any  power  of  a  number  is  equal  to  the 
logarithm  of  the  number  multiplied  by  the  exponent  of  the 
power. 

Let      (1)     b*  =n;    then,  by  def.,  log  n  =x. 
(1)*:=(2)     bpx=np;   then,  by  def.,  log  np=px. 

.  ' .     log  n ''  =  p  log  n. 

23.   Rule. 

1.  Find   the  logarithm  of  the  number  and  multiply  it  by 
the  exponent  of  the  power,  and  the  product  will  be  the  loga- 
rithm of  the  power. 

2.  Find  the  number^  corresponding  which  will  be  the  power. 

24:.   Examples. 

1.  Find  the  cube  of  .034. 

(1)  log  .034  =  2T53148 
(1)  X  3  =  (2)  log  .034^  5.59444    .  • .  .0343—  .000039305. 

2.  Find  the  square  of  25.7.  Ans.  660.47. 


22  LOGARITHMS. 

3.  Find  the  fourth  power  of  .75.          Ans.  .3164. 

4.  Find  the  cube  of  8.07.  Ans.  525.55. 

5.  Find  the  fifth  power  of  .9.  Ans.  .59047. 

EVOLUTION   BY   LOGARITHMS. 
25.    Proposition. 

The  logarithm  of  any   root   of  a  number  is  equal  to  the 
logarithm  of  the  number  divided  by  the  index  of  the  root. 

Let  (1)     bx  =  n;  then,  by  def.,  log  n  =  x. 

1/'(1)  =  (2)     brr  =  ir/n/   then,  by  def.,  log  V^n  =  ~ 

r,-       log  n 
.  • .     log  V  n  =  — - — . 

26.   Rule. 

1.  Find    the   logarithm  of   the   number,  divide  it  by  the 
index  of  the  root,  and   the  quotient   will    be   the  logarithm 
of  the  root. 

2.  Find  the  number  corresponding  which  will  be  the  root. 

27.    Examples. 

1.  Extract  the  square  root  of  .75. 

(1)     log      .75=T.87506 

(1)  -s-  2  =  (2)    log  V  J5  =T93753     .  * .   VTft  =  .86602. 
Scholium.     L87506  ~  2  =  (2  +  1.87506)  ~-  2  =  T93753. 

2.  Extract  the  cube  root  of  91125.  Ans.  45. 

3.  Find  the  value  of  £  i/67  -4ns.  .89443. 

4.  Extract  the  fifth  root  of  .075.  Ans.  .59569. 


5.  Find  the  value  of  3  /B7.5  X  Q78)2        Ans   >676317. 

12.5X5.9 


PLANE  TRIGONOMETRY.  23 


TRIGONOMETRY. 

28.   Definition  and  Classification. 

Trigonometry  is  that  branch  of  Mathematics  which 
treats  of  the  solution  of  triangles. 

Trigonometry  is  divided  into  two  branches  —  Plane 
and  Spherical. 


PLANE   TRIGONOMETRY. 
29.   Definition. 

Plane  Trigonometry  is  that  branch  of  Trigonometry 
which  treats  of  the  solution  of  plane  triangles. 

30.   Parts  of  a  Triangle. 

Every  triangle  has  six  parts — three  sides  and  three 
angles. 

If  three  parts  are  given,  one  being  a  side,  the  re- 
maining parts  can  be  computed. 

If  the  three  angles  only  are  given,  the  triangle  is 
indeterminate,  since  an  infinite  number  of  similar 
triangles  will  satisfy  the  conditions. 

31.    Sexagesimal  Division  of  Angles  and  Arcs. 

The  horizontal  diameter,  0  P,  called  the  primary  di- 
ameter,   and   the    vertical    diameter, 
0'  P',  called  the  secondary  diameter, 
divide  the  circumference   into  four 
equal  parts,  called  quadrants. 

0  (7  is  the  first  quadrant,  0'  P  the 
second,  P  P'  the  third,  and  P'  0  the 
fourth. 


24  TRIGONOMETRY. 

A  degree  is  one-ninetieth  of  a   right   angle,  or  of  a 

quadrant.  • 

A  minute  is  one-sixtieth  of  a  degree. 

A  second  is  one-sixtieth  of  a  minute. 

Thus,  25°  34'  46"  denote  25  degrees,  34  minutes,  and 
46  seconds. 

An  angle,  whose  vertex  is  at  the  center,  has  the 
same  numerical  measure,  or  contains  the  same  number 
of  degrees,  minutes,  and  seconds,  as  the  arc  of  the 
circumference  intercepted  by  its  sides. 

32.   Centesimal  Division  of  Angles  and  Arcs. 

A  grade  is  one-hundreth  of  a  right-angle,  or  of  a 
quadrant. 

A  minute  is  one-hundreth  of  a  grade. 

A  second  is  one-hundreth  of  a  minute. 

Thus,  I9  24'  40"  denotes  7  grades,  24  minutes,  and 
40  seconds. 

Z9~'        =27~1          =^8T' 

i!     r_?r     1W__8T 
"10"'       ~50'         ~250' 

Let  d,  m,  s,  respectively,  denote  an  angle  expressed 
in  degrees,  sexagesimal  minutes  and  seconds,  and  let' 
g,  M,  <r,  respectively,  denote  the  same  angle  expressed 
in  grades,  centesimal  minutes  and  seconds,  then  ex- 
pressing the  ratio  of  the  angle  to  a  right  angle  in 
each  kind  of  units,  we  shall  have : 

damn  s  * 


90  "100    5400~  10000'   324000  ~  1000000 
9        27        81 


10        50        250 


PLANE  TRIGONOMETRY.  25 

Let  r  denote  the  radius,  and  71=3.14159265358979... 
T:  r  =  a  semi-circumference  =  180°  ==  200"  =  two  right 
angles. 

^j-r  =  &  quadrant  PR  90°  =  3  1007  f==  one  right  angle. 

2  -  r  —  a  circumference  —  360°  :  =  400"  =  four   right, 
angles. 

If  r—1,  the  above  expressions  become,  respectively, 


33.    Unit  of  Circular  Measure. 

The  unit  of  circular  measure  is  that  angle  at  the 
center  whose  intercepted  arc  is  equal  in  length  to  the 
radius. 

Let  u  denote  the  unit  of  circular  measure,  and  r  the 
radius. 

Then,  since  *  r  =  the  semi-circumference,  -KU  —  180° 
==  200'. 

-j  CAO  900  •' 

u  =  —  -=  57°.  29577951  ..  =  -    -=63*.  6619772... 

77  7T 

Let    rf,    g,    Cj  respectively,    denote    the    number    of 

degrees,  grades,  and   units  of  circular   measure   in   an 
angle;   then, 

180  200  ~  TT 

:__C)  9=—c,      C^l80rf'    e=WOg' 

34.    Origin,  Termini  and  Situation  of  Arcs. 

The  origin  of  an  arc  is  the  extremity  at  which  it 
begins. 

The  primary  origin  of  arcs  is  at  the  right  extremity 
of  the  primary  diameter. 

The  secondary  origin  of  arcs  is  at  the  upper  extremity 
of  the  vertical  diameter. 
S.  N.  3. 


26  TRIG  OXOMETR  Y. 

The  terminus  of  an  arc  is  the  extremity  at  which 
it  ends. 

An  arc  is  said  to  be  situated  in 
that  quadrant  in  which  its  ter- 
minus is  situated,  thus : 

The  arc  OT  is  in  the  first  quad- 
rant. 

The  arc  00'  T'  is  in  the  second 
quadrant. 

The  arc  OPT"  is  in  the  third  quadrant. 
The  arc  OPT'"  is  in  the  fourth  quadrant, 

35.  Positive  and  Negative  Arcs. 

Positive  arcs  are  those  which  are  estimated  in  the 
direction  contrary  to  that  of  the  motion  of  the  hands 
of  a  watch. 

Negative  arcs  are  those  which  are  estimated  in 
the  same  direction  as  that  of  the  motion  of  the  hands 
of  a  watch. 

Thus,  OT,  OT',  OT",  OT'",  estimated  to  the  left, 
are  positive,  and  OT'",  OT",  OT',  OT,  estimated  to 
the  right,  are  negative. 

36.  The  Complement  of  an  Arc. 

The  complement  of  an  arc  or  angle  is  90°  minus  that 
arc  or  angle. 

If  the  arc  or  angle  is  less  than  90°,  its  comple- 
ment is  positive. 

If  the  arc  or  angle  is  greater  than  90°,  its  comple-- 
ment  is  negative. 

The  complement  of  an  arc,  geometrically  considered, 
is  the  arc  estimated  from  the  terminus  of  the  given 
arc  to  the  secondary  origin.  Therefore,  by  the  preced- 
ing article,  the  complement  of  an  arc  will  be  positive 


FUNCTIONS.  27 

or    negative,    according   as   the   arc    is    less   or    greater 
than  90°. 

TO'  is  the  complement  of  OT,  and  is  positive. 

TO'  is  the  complement  of  OT',  and  is  negative. 

T"O'  is  the  complement  of  OT",  and  is  negative. 

T'"O'  is  the  complement  of  OT'",  and  is  negative. 

37.   The  Supplement  of  an  Arc. 

The  supplement  of  an  arc  or  angle  is  180°  minus 
that  arc  or  angle. 

If  the  arc  or  angle  is  less  than  180°,  its  supple- 
ment is  positive. 

If  the  arc  or  angle  is  greater  than  180°,  its  supple- 
ment is  negative. 

The  supplement  of  an  arc,  geometrically  considered, 
is  the  arc  estimated  from  the  terminus  of  the  given 
arc  to  the  left-hand  extremity  of  the  primary  diameter. 
Therefore,  by  article  35,  the  supplement  of  an  arc  will 
be  positive  or  negative,  according  as  the  arc  is  less  or 
greater  than  180°. 

TP  is  the  supplement  of  OT,  and  is  positive. 

T'P  is  the  supplement  of  OT',  and  is  positive. 

T"P  is  the  supplement  of  OT",  and  is  negative. 

T"fP  is  the  supplement  of  OT"",  and  is  negative. 

TKIGONOMETKICAL  FUNCTIONS. 

38.   Preliminary  Definitions  and  Remarks. 

1.  A  function  of  a  quantity  is  a  quantity  whose  value 
depends  on  the  given  quantity. 

2.  The  trigonometrical  functions,   called  also   circular 
functions,  are   auxiliary  lines,  which    are   functions   of 
an  arc  or  of  the  angle  which  has  the  same  measure  as 
that  arc. 


28  TRIG  OSOMETR  Y. 

3.  These    functions    are    eight    in    number,   and    are 
called   the  sine,  co-sine,  versed-sine,   co-ver sect-sine,   tangent, 
co-tangent,  secant   and   co-secant,  which    are    abbreviated 
thus,  sin,  cos,  vers,  covers,  tan,  cot,  sec,  cosec. 

4.  The  solution  of  triangles  is  accomplished  by  the 
aid  of  these  functions,  since  they  enable  us  to  ascertain 
the  relations  which  exist  between  the  sides  and  angles 
of  triangles. 

5.  The   primary  origin  will  be  taken  as  the  common 
origin  of  the  arcs,  unless  the  contrary  is  stated. 

6.  The  origin  of  any  arc,  wherever  situated,  may  be 
considered  the  primary  origin  of  that  arc ;   and  its  sec- 
ondary origin  is  a  quadrant's  distance  from  the  primary 
origin,  in  the  direction  of  the  positive  or  negative  arcs, 
according  as  the  given  arc  is  positive  or  negative. 

7.  An  arc  will  be  considered  positive  unless  the  con- 
trary is  stated. 

8.  The  primary  diameter  passes  through  the  primary 
origin ;    and   the  secondary  diameter,  through  the  sec- 
ondary origin. 

9.  Lines  estimated   upward,  toward   the  right,  or  from 
the  center   toward   the   terminus  of  the   arc,  are  considered 
positive. 

10.  Lines  estimated  downward,  toward  the  left,  or  from, 
the   center   and    the    terminus   of    the    arc,    are    considered 
negative. 

11.  The  limiting  values  of  the  circular  functions  are 
their  values  for  the  arcs  0°,  90°,  180°,  270°,  360°. 

12.  The   sign  of  a  varying   quantity,  up  to  a  limit, 
is  its  sign  at  the  limit. 

13.  Point  out  positive  arcs  in  the  following  diagram, 
and  the  origin  and  terminus  of  each. 

14.  Point  out  negative  arcs,  the  origin,  terminus  and 
primary  diameter  of  each. 

15.  Point  out  the  positive  lines,  also  the  negative. 


FUNCTIONS. 


39.  The  Sine  of  an  Arc. 

The  sine  of  an  arc  is  the  perpendicular  distance  of  its 
terminus  from  the  primary  diameter. 

MT  is  the  sine  of  the  arc  OT. 

M'T'  is  the  sine  or  the  arc  OT'. 

M'T"  is  the  sine  of  the  arc  OT". 

MT'"   is   the  sine  of  the  arc  OT'". 

By  the  arcs  OT"  and  OT'",  we  are 
to  understand  the  positive  arcs,  and 
not  the  negative  arcs  designated  by 
the  same  letters. 

The  sine  of  an  arc  is  the  sine  of  the  angle  measured 
by  that  arc. 

Thus,  MT,  the  sine  of  the  arc  OT,  is  the  sine  of 
the  angle  OCT,  which  is  measured  by  the  arc  OT; 
and  similarly  for  the  other  arcs  and  angles. 

The  arcs  OT  and  OT'  are  in  the  first  and  second 
quadrants,  respectively,  and  their  sines  MT  and 
M'T'  are  estimated  upward,  and  are  therefore  positive; 
hence, 

The  sine  of  an  arc  in  the  first  or  second  quadrant  is 
positive. 

The  arcs  OT"  and  OT'"  are  in  the  third  and 
fourth  quadrants,  respectively,  and  their  sines,  M'  T" 
and  MT'",  are  estimated  downward,  and  are  there- 
fore negative;  hence, 

The  sine  of  an  arc  in  the  third  or  fourth  quadrant  is 
negative. 

Let  the  chord  TT'  be  parallel  to  the  primary  diame- 
ter OP,  then  will  M'  T'  be  equal  to  MT,  and  the  arc 
OT  will  be  equal  to  the  arc  T'  P;  but  the  arc  T' P 
is  the  supplement  of  the  arc  OT';  therefore,  the  arc 
OT  is  the  supplement  of  the  arc  OT';  but  M'.T', 


30  TRIGONOMETRY. 

the  sine  of  the  arc  0T',  is  equal  to  MTJ  the  sine  oC 
the  arc  OT,  the  supplement  of  OT';  hence, 

The  sine  of  an  arc  is  equal  to  the  sine  of  its  supplement. 

The  sine  of  0°  is  0.  As  the  arc  increases  from  0° 
to  90°,  the  sine  increases  from  0  to  +1.  As  the  arc 
increases  from  90°  to  180°,  the  sine  decreases  from  ~f  1 
to  -fO.'  As  the  arc  increases  from  180°  to  270°,  the 
sine  passes  through  0,  changes  its  sign  from  -f  to  — , 
and  increases  numerically,  but  decreases  algebraically 
from  —  0  to  —  1.  As  the  arc  increases  from  270°  to 
360°,  the  sine  decreases  numerically,  but  increases  al- 
gebraically from  —  1  to  —  0. 

Hence,  for  the  limiting  values  of  the  sine,  we  have 
sin  0°  =  0,  sin  90°  =  -f  1,  sin  180°  =.  -f  0, 

sin  270°  =  —  1,  sin  360°  =  —  0. 

40.   The  Co-sine  of  an  Arc. 

The  co-sine  of  an  arc  is  the  perpendicular  distance 
of  its  terminus  from  the  secondary  diameter. 

NT  is  the  co-sine  of  the  arc  OT. 
NT'  is  the  co-sine  of  the  arc  OT'. 
N'T"  is  the  cosine  of  the  arc  OT". 
N'T'"  is  the  co-sine  of  the  arc  OT'". 
The   arcs  OT  and  OT'"  are    in   the 
first  and  fourth  quadrants,  respective- 
ly, and   their  co-sines  NT  and  N'T'" 
are  estimated  toward  the  right,  and  are  therefore  posi- 
tive; hence, 

The  co-sine  of  an  arc  in  the  first  or  fourth  quadrant  -is 
positive. 

The  arcs  OT'  and  OT"  are  in  the  second  and  third 
quadrants,  respectively,  and  their  co-sines,  NT'  and 
JVT",  are  estimated  toward  the  left,  and  are  therefore 
negative;  hence, 


FUNCTIONS.  31 

The  co-sine  of  an  arc  in  the  second  or  third  quadrant  is 
negative. 

The  word  co-sine  is  an  abbreviation  of  complementi 
sinus,  the  sine  of  the  complement.  In  fact,  NT,  the 
co-sine  of  OT,  is  the  sine  of  O'T,  the  complement  of 
OT;  hence, 

The  co-sine  of  an  arc  is  the  sine  of  its  complement. 

MT,  the  sine  of  OT,  is  the  co-sine  of  O'T,  the  com- 
plement of  OT;  hence, 

The  sine  of  an  arc  is  the  co-sine  of  its  complement. 

Since  the  radius  CO'  is  perpendicular  to  the  chord 
TT'j  NT  and  NT'  are  numerically  equal;  but  since 
NT  is  estimated  toward  the  right,  and  NTf  toward 
the  left,  they  have  contrary  signs;  hence,  NT=  —  NT'; 
but  NT  is  the  co-sine  of  OT,  and  NT'  is  the  co-sine 
of  OT',  the  supplement  of  OT;  hence, 

The  co-sine  of  an  arc  is  equal  to  minus  the  co-sine  of  its 
supplement. 

It  is  evident  that  CN  is  equal  to  the  sine  of  OT, 
or  of  OT',  and  that  CN'  is  equal  to  the  sine  of  OT", 
or  of  OT'";  hence, 

The  sine  of  an  arc  is  equal  to  that  part  of  the  secondary 
diameter  from  the  center  to  the  foot  of  the  co-sine. 

It  is  evident  that  CM  is  equal  to  the  co-sine  of  OT, 
or  of  OT'",  and  that  CM'  is  equal  to  the  co-sine  of 
OT'  or  of  OT";  hence, 

The  co-sine  of  an  arc  is  equal  to  that  part  of  the  primary 
diameter  from  the  center  to  the  foot  of  the  sine. 
;      The  co-sine  of  0°  is  -f  1.     As  the  arc  increases  from 
0°  to  90°,  the  co-sine  decreases  from  -}-  1  to  -f  0.     As 
the  arc  increases   from   90°  to  180°,  the   co-sine   passes 
through  0,  changes  its  sign  from  -f-  to  — ,  and  increases 
numerically,   but   decreases    algebraically  from    --  0   to 
-  1.     As   the   arc   increases  from  180°  to  270°,  the  co- 
sine decreases  numerically,  but   increases  algebraically 


32  TRIG  OSOMETR  Y. 

from  ---1  to  —  0.     As  the   arc   increases   from   270°  to 
360°,  the  co-sine  passes  through  0,  changes  its  sign  from 
-  to  -f,  and  increases  from  -|-  0  to  +  1. 
Hence,  for  the  limiting  values  of  the  co-sine,  we  have 
cos  0°  =:  +  1,          cos  90°  =  +  0,  cos  180°  =  —  1, 

cos  270°  =  —  0,       cos  360°  ==  +  1. 

41.   The  Versed-Sine  of  an  Arc. 

The  versed-sine  of  an  arc   is  the   perpendicular  dis- 
tance   of   the    primary   origin    from 
the  sine. 

MO  is  the  versed-sine  of   the   arc 
OT,  and  of  the  arc  OT'". 

M'O  is  the  versed-sine  of  the  arc 
OT',  and  of  the  arc  OT". 

The  versed-sine  of  an  arc,  in   any 
quadrant,  is   estimated   to   the    right,  and    is    therefore 
positive ;   hence, 

The  versed-sine  is  always  positive. 

The  versed-sine  of  0°  is  0.  As  the  arc  increases  from 
0°  to  90°,  the  versed-sine  increases  from  0  to  -+-  1.  As 
the  arc  increases  from  90°  to  180°,  the  versed-sine  in- 
creases from  -f- 1  to  +  2.  As  the  arc  increases  from  180° 
to  270°,  the  versed-sine  decreases  from  -f-  2  to  -f  1-  As 
the  arc  increases  from  270°  to  360°,  the  versed-sine 
decreases  from  -f  1  to  -f-  0. 

Hence,  the  limiting  values  of  the  versed-sine  are 
vers  0°  =0,  vers  90°  ==4r  1,  vers  180°  ==  +  2, 

vers  270°  ==«--£!,         vers  360°  =:  -f  0. 

What  are  the  least  and  greatest  values  of  the  sine, 
and  what  are  the  corresponding  arcs? 

What  are  the  least  and  greatest  values  of  the  co-sine, 
and  what  are  the  correspond!  17  g  arcs? 

What  are  the  least  and  greatest  values  of  the  versed- 
sine,  and  what  are  the  corresponding  arcs? 


FUNCTIONS.  33 

42,   The  Co-versed-sine  of  an  Arc. 

The  co-versed-sine  of  an  arc  is  the  perpendicular  dis- 
tance of  the  secondary  origin  from  the  co-sine. 

Thus,  see  diagram  of  the  last  article,  NO'  is  the  co- 
versed-sine  of  the  arc  OT,  and  of  the  arc  OT';  N'O' 
is  the  co-versed-sine  of  the  arc  OT",  and  of  the  arc 
OT". 

The  co-versed-sine  of  an  arc  in  any  quadrant  is  esti- 
mated upward,  and  is  therefore  positive;  hence, 

The  co-versed-sine  is  always  positive. 

The  word  co-versed-sine  is  an  abbreviation  of  comple- 
menti  vcrsatus  sinus,  the  versed  or  turned  sine  of  the  com- 
plement. In  fact,  NO',  the  co-versed-sine  of  OT,  is  the 
versed-sine  of  O'T,  the  complement  of  OT;  hence, 

The  co-versed-sine  of  an  arc  is  the  versed-sine  of  its  com- 
plement. 

MO,  the  versed-sine  of  OT,  is  the  co-versed-sine  of 
O'T,  the  complement  of  OT;  hence, 

The  versed-sine  of  an  arc  is  the  co-versed-sinc  of  its  com- 
plement. 

The  co-versed-sine  of  0°  is  1.  As  the  arc  increases 
from  0°  to  90°,  the  co-versed-sine  decreases  from  -j-  1  to 
}  0.  As  the  arc  increases  from  90°  to  180°,  the  co- 
versed-sine  increases  from  -f  0  to  -f  1.  As  the  arc  in- 
creases from  180°  to  270°,  the  co-versed-sine  increases 
from  -f  1  to  -f  2.  As  the  arc  increases  from  270° 
to  360°,  the  co-versed-sine  decreases  from  -f  2  to  + 1. 
Hence,  the  limiting  values  of  the  co-versed-sine  are, 
covers  0°  ==  -f  1,  covers  90°  ===  -f  0,  covers  180°  ==  -f  1, 
covers  270°  =  -f  2,  covers  360°  ==  -f  1. 

What  are  the  least  and  greatest  values  of  the  co- 
versed-sine,  and  what  are  the  corresponding  arcs? 

Trace  the  arcs  from  0°  to  360°,  and  the  changing 
functions. 


34  TRIGONOMETRY. 

43.   The  Tangent  of  an  Arc. 

The  tangent  of  an  arc  is  the  perpendicular  to  the 
primary  diameter,  produced  from  the  primary  origin, 
till  it  meets  the  prolongation  of  the  diameter  through 
the  terminus  of  the  arc. 

OR  is  the   tangent  of  the  arcs  OT 
and  OT". 

OR  is  the  tangent  of  the  arcs  OT' 
and  OT'". 

The  arcs  OT  and  OT"  are  in  the 
first  and  third  quadrants,  respectively, 
and   their   tangent,   OR,    is   estimated   upward,  and    is 
therefore  positive;   hence, 

The  tangent  of  an  arc  in  the  first  or  third  quadrant  is- 
positive. 

The  arcs  OT'  and  OT'"  are  in  the  second  and  fourth 
quadrants,  respectively,  and  their  tangent,  OR',  is  es- 
timated downward,  and  is  therefore  negative;  hence, 

The  tangent  of  an  arc  in  the  second  or  fourth  quadrant  is 
negative. 

Let  the  arc  OT7  be  equal  to  the  arc  T'P.  Then, 
since  T'P  is  the  supplement  of  OT',  OT  will  be  the 
supplement  of  OT';  but  the  arc  T'"0  is  the  sup- 
plement of  OT7';  hence,  OT=T'"0,  and  the  angle 
OCT  is  equal  to  the  angle  OCT'".  The  angle  COR 
is  equal  to  the  angle  COR',  since  each  is  a  right 
angle.  Hence,  the  two  triangles  COR  and  COR  have 
two  angles,  and  the  included  side  of  the  one  equal 
to  two  angles  and  the  included  side  of  the  other,  each 
to  each,  and  are  therefore  equal  in  all  their  parts. 
Hence,  OR,  opposite  the  angle  OCR,  is  equal  to  OR, 
opposite  the  equal  angle  OCR.  Since  OR  is  esti- 
mated upward,  and  OR'  downward,  they  have  contrary 
signs ;  hence,  OR  -  -  OR.  But  OR  is  the  tangent 


FUNCTIONS.  35 

of  the  arc  OT,  and  OR  is  the  tangent  of  the  arc  07", 
the  supplement  of  OT;   hence, 

The  tangent  of  an  arc  is  equal  to  minus  the  tangent  of 
its  supplement. 

The  tangent  of  0°  is  0.  As  the  arc  increases  from 
0°  to  90°,  the  tangent  increases  from  0  to  -f  oo.  As 
the  arc  increases  from  90°  to  180°,  the  tangent  passes 
through  oc,  changes  its  sign  from  -f-  to  — ,  and  de: 
creases  numerically,  but  increases  algebraically  from 

-oo  to  —  0.      As  the  arc  increases  from  180°  to  270°, 
the   tangent   passes   through   0,    changes  its  sign  from 

-  to  -f ,  and  increases  from  -j-  0  to  -f  oc.  As  the  arc 
increases  from  270°  to  360°,  the  tangent  passes  through 
oo,  changes  its  sign  from  -|-  to  — ,  and  decreases  nu- 
merically, but  increases  algebraically  from  —  oo  to  —  0. 
Hence,  for  the  limiting  values  of  the  tangent  we  have 
tan  0°  =  0,  tan  90°  =  :  -f  oo,  tan  180°  =  —  0, 
tan  270°  =  -f  oo,  tan  360°  =  —  0. 

44.   The  Co-tangent  of  an  Arc. 

The  jo-tangent  of  an  arc  is  the  perpendicular  to  the 
secondary  diameter,  produced  from  the  secondary  origin, 
till  it  meets  the  prolongation  of  the  diameter  through 
the  terminus  of  the  arc. 

O^S  is  the  co-tangent  of  OT  and  OT". 

O'S'  is  the  co-tangent  of  OT'  and  OT". 

The  arcs  OT  and  OT"  are  in  the  first  and  third 
quadrants,  respectively,  and  their  co-tangent,  0'$,  is 
estimated  to  the  right,  and  is  therefore  positive;  hence, 

The  co-tangent  of  an  arc  in  the  first  or  third  quadrant 
is  positive. 

The  arcs  OT'  and  OT'"  are  in  the  second  and  fourth 
quadrants,  respectively,  and  their  co-tangent,  O'S',  is  es- 
timated to  the  left,  and  is  therefore  negative;  hence, 


86  TRIG  OSOMETR  Y. 

Tlic  co-tangent  of  an  arc  in  the  second  or  fourth  quadrant 
is  negative. 

The  word  co-tangent  is  an  abbreviation  of  complementi 
tangens,  the  tangent  of  the  complement.  In  fact,  O'S, 
the  co-tangent  of  OT,  is  the  tangent  of  O'T,  the  com- 
plement of  OT;  hence, 

The  co-tangent  of  an  arc  'is  the  tangent  of  its  complement. 

OR,  the  tangent  of  OT,  is  the  co-tangent  of  O'T,  the 
complement  of  OT;  hence, 

The  tangent  of  an  arc  is  the  co-tangent  of  its  complement. 

Let  the  arcs  OT  and  T'P  be  equal.  Then,  since 
T'P  is  the  supplement  of  OT',  OT  will  be  the  supple- 
ment of  OT'. 

The  arcs  O'T  and  O'T'  are  equal,  since  they  are 
complements  of  the  equal  arcs  OT  and  T'P;  hence, 
the  angles  O'CT  and  O'CT',  measured  by  these  equal 
arcs,  are  equal.  The  angles  CO'S  and  GO'S'  are  equal, 
since  each  is  a  right  angle.  Hence,  the  two  triangles 
CO'S  and  COS'  have  the  common  side  CO',  and  the 
two  adjacent  angles  equal,  and  are  therefore  equal  in 
all  their  parts;  and  O'S,  opposite  the  angle  O'CS,  is 
equal  to  O'S',  opposite  the  equal  angle  O'CS'. 

Since  O'S  is  estimated  to  the  right,  and  O'S'  to  the 
left,  they  have  contrary  signs;  hence,  O'S  =  -O'S'. 
But  O'S  is  the  co-tangent  of  OT,  and  O'S'  is  the  co- 
tangent of  OT',  the  supplement  of  OT;  hence, 

The  co-tangent  of  an  arc  is  equal  to  minus  the  co-tangent  of 
its  supplement. 

The  co-tangent  of  0°  is  -foe.  As  the  arc  increases 
from  0°  to  90°,  the  co-tangent  decreases  from  -f  GO  to 
-J-  0.  As  the  arc  increases  from  90°  to  180°,  the  co- 
tangent passes  through  0,  changes  its  sign  from  -f  to 
— ,  and  increases  numerically,  but  decreases  algebra- 
ically from  —0  to  --  x.  As  the  arc  increases  from 
180°  to  270°,  the  co-tangent  passes  through  cc,  changes 


FUNCTIONS. 


37 


its  sign  from  —  to  -f?  and  decreases  from  -f-  oo  to  -fO. 
As  the  arc  increases  from  270°  to  360°,  the  co-tangent 
passes  through  0,  changes  its  sign  from  -j-  to  — ,  and 
increases  numerically,  but  decreases  algebraically  from 
-0  to  —oo. 

Hence,  the  limiting  values  of  the  co-tangent  are 
cot  0°  =  -f-  oo,  cot  90°  s=  -f  0,  cot  180°  =  —  oo, 
cot  270°  =  4-0,  cot  360°  —  —  oo. 


4:5.   The  Secant  of  an  Arc. 

The  secant  of  an  arc  is  the  line  drawn  from  the  center 
of.  the  circle  to  the  terminus  of .  the 
tangent. 

CR  is  the  secant  of  OT  and  OT". 

CR  is  the  secant  of  OT'  and  OT'". 

The  arcs  OT  and  OT'"  are  in  the 
first   and   fourth    quadrants,    respect- 
ively,   and    their    secants,    CR    and 
CR'  are  estimated  from  the  center  toward  the  termini 
of  the  arcs,  and  are  therefore  positive ;    hence, 

The  secant  of  an  arc  in  the  first  or  .fourth  quadrant  is 
positive. 

-The  arcs  OT'  and  OT"  are  in  the  second  and  third 
quadrants,  respectively,  and  their  secants,  CR'  and  CR, 
are  estimated  from  the  center,  from  the  termini  of  the 
arcs,  and  are  therefore  negative;  hence, 

The  secant  of  an  arc  in  the  second  or  third  quadrant  is 
negative. 

Let  the  arcs  OT  and  T'P  be  equal.  Then,  since 
T'P  is  the  supplement  of  OT',  OT  is  the  supplement 
of  OT';  but  T'"0  is  the  supplement  of  OT';  therefore, 
T'"0  is  equal  to  OT,  and  the  angle  T'"CO,  measured 
by  T'"0,  is  equal  to  the  angle  OCT,  measured  by  the 
equal  arc  OT.  The  right  angles  COR  and  COR'  are 


38 


TRIGONOMETRY. 


equal.  Hence,  in  the  triangles  having  the  common 
side  CO,  and  the  two  adjacent  angles  equal,  CR  is 
equal  to  CR';  but  CR,  the  secant  of  OT,  is  positive; 
and  CR',  the  secant  of  OT',  the  supplement  of  OT,  is 
negative;  hence,  CR=  —CR;  hence, 

The  secant  of  an  arc  is  equal  to  minus  the  secant  of  its 
supplement, 

The  secant  of  0°  is  -f  1.  As  the  arc  increases  from 
0°  to  90°,  the  secant  increases  from  -}-  1  to  -j-  oo.  As 
the  arc  increases  from  90°  to  180°,^  the  secant  passes 
through  GO,  changes  its  sign  from  -f  to  — ,  and  de- 
creases numerically,  but  increases  algebraically  from 
—  oo  to  —  1.  As  the  arc  increases  from  180°  to  270°, 
the  secant  increases  numerically,  but  decreases  alge- 
braically from  —  1  to  —  oo.  As  the  arc  increases  from 
270°  to  360°,  the  secant  passes  through  oo,  changes  its 
sign  'from  --  to  -j-,  and  decreases  from  -f  oo  to  -J- 1. 
Hence,  for  the  limiting  values  of  the  secant  we  have 
sec  0°  P=  -f  1,  sec  90°  fc:  -f  oo,  sec  180°  =  —  1, 

sec  270°  =  —  oo,          sec  360°  =  -f  1. 


46.   The  Co-secant  of  an  Arc. 

The  co-secant   of  an   arc  is  the  line  drawn   from  the 
center  of  the  circle  to  the   terminus 
of  the  co-tangent. 

CS  is  the  co-secant  of  OT  and  OT". 

CS'  is  the  co-secant  of  OT'  and  OT'". 

The   arcs  OT  and  OT'  are  in  the 
first    and   second   quadrants,   respect- 
ively, and  their  co-secants  CS  and  CS' 
are   estimated  from   the   center   toward  the  termini  of 
the  arcs,  and  are  therefore  positive;  hence, 

The  co-secant  of  an  arc  in  the  first  or  second  quadrant 
is  positive. 


FUNCTIONS.  39 

The  arcs  OT"  and  OT'"  are  in  the  third  and  fourth 
quadrants,  respectively,  and  their  co-secants,  CS  and 
CS',  are  estimated  from  the  center  and  the  termini  of 
the  arcs,  and  are  therefore  negative;  hence, 

The  co-secant  of  an  arc  in  the  third  or  fourth  quadrant 
is  negative. 

The  word  co-secant  is  an  abbreviation  of  complement 
secans,  the  secant  of  the  complement.  In  fact,  CS,  the 
co-secant  of  OT,  is  the  secant  of  O'T,  the  complement 
of  OT;  hence, 

The  co-secant  of  an  arc  is  the  secant  of  its  complement. 

CR,  the  secant  of  OT,  is  the  co-secant  of  O'T,  the 
complement  of  OT;  hence, 

The  secant  of  an  arc  is  the  co-secant  of  its  complement. 

Let  the  arcs  OT  and  T'P  be  equal.  Then,  since  T'P 
is  the  supplement  of  OT',  OT  will  be  the  supplement 
of  OT'.  0'T  =  O'T',  since  they  are  complements  of 
equal  arcs.  Hence,  the  angle  O'CT,  measured  by  the 
arc  O'T,  is  equal  to  the  angle  OCT',  measured  by 
the  equal  arc  O'T'.  The  right  angles,  COS  and  COS', 
are  equal. 

Hence,  in  the  triangles  having  the  common  side  CO', 
and  the  two  adjacent  angles  equal,  CS  is  equal  to 
CS';  but  CS  is  the  co-secant  of  OT,  and  positive,  and 
CS'  is  the  co-secant  of  OT',  and  positive;  hence, 

The  co-secant  of  an  arc  is  equal  to  the  co-secant  of  its 
supplement. 

The  co-secant  of  0°  is  -j-  oo.  As  the  arc  increases 
from  0°  to  90°,  the  co-secant  decreases  from  -|-  oo  to  -f-1. 
As  the  arc  increases  from  90°  to  180°,  the  co-secant  in- 
creases from  +  1  to  -f-  oo.  As  the  arc  increases  from 
180°  to  270°,  the  co-secant  passes  through  oo,  changes 
its  sign  from  +  to  — ,  and  decreases  numerically,  but 
increases  algebraically  from  —  oo  to  —  1.  As  the  arc 
increases  from  270°  to  360°,  the  co-secant  increases 


40 


TRIGONOMETRY. 


numerically,  but  decreases  algebraically  from  — 1  to 
—  oo.  Hence,  the  limiting  values  of  the  co-secant  are 
cosec  0°  =  4-  oo,  cosec  90°  ==  4  1,  cosec  180°  =;  -f  ex, 
cosec  270°  =  —  1,  cosec  360°  =  -  oo. 

To  aid  the  memory,  and  for  convenience  of  reference, 
we  give  the  following  tabular  summaries: 


47.   Signs  of  the  Circular  Functions. 


Functions. 

Istq. 

2d  q. 

Uq. 

4th  q. 

sine. 

4 

4 

— 

— 

co-sine. 

+ 

— 

— 

4 

versed-sine. 

4 

4 

4 

+ 

co-  versed-sine. 

4- 

-  + 

t 

+ 

tangent. 

4 

— 

+ 

— 

co-tangent. 

4 

— 

4 

—  - 

secant. 

4 

— 

— 

H- 

co-secant. 

4 

Ht 

— 

— 

48.  Limiting  Tallies  of  the  Circular  Functions. 


0° 

90° 

180° 

270° 

360° 

sin  =  +  0 

sin  -=4  1 

sin  ==40 

sin  =  —  1 

sin  =    -0 

cos  —  -f-  1 

cos  -  =  40 

cos   =     -  1 

cos  =  —  0 

cos  =4  "A 

vsin—  -{-  0 

vsin  —  4  1 

vsin=4-2 

vsin—  4  1 

vsin=r40 

cvs  =^  -j-  1 

cvs  -=40 

cvs  =41 

cvs  =42 

cvs  =4  1 

tan  =  +  0 

tan  =4  oc 

tan  =—  0 

tan  ==4~.  °° 

tan  =    -0 

cot  =  4  °° 

cot  =  =  40 

cot  =  —  oo 

cot   =+0 

cot  =  —  co 

sec  =  +  1 

sec  =4~  °°'  sec  =~~  1 

sec  =  —  oo 

sec  =4  A 

cose  =  -f  oo  cose  =^4  1  1  cose=4  oc!  cose  =  -  1 

cose=  —  oo 

NATURAL   FUNCTIONS.  41 

49.    Problem. 

To  find  any  function  of  an  angle  to  the  radius  R,  in 
terms  of  the  corresponding  function  of  the  same  angle  to  the 
radius  1,  and  the  reverse. 

Let  sin  Ol  denote  sin  C  to 
the  radius  CT=l,  and  sin  CR 
denote  sin  C  to  the  radius 
CT'  =  R. 

From   similar  triangles, 

CT    :    CT'   :  :   MT      :   M'T', 

or  1       :    R       :  :   sin  Cl  :    sin  OR. 

.-.(I)     sinCJI  =  sinC'1X^.       -  '  •  (2)    sin 


Let  formulas  for  other  functions  be  deduced;  hence, 

1.  Any  function  of  an  angle  to  the  radius  R  is  equal  to 
the  corresponding  function   of  the  same  angle  to  the  radius 
1,  multiplied  by  R. 

2.  Any  function  of  an  angle  to  the  radius  1  is  equal  to 
the  corresponding  function  of  the  same  angle  to  the  radius 
R,  divided  by  R. 


TABLE   OF  NATURAL   FUNCTIONS. 
50.   Description  of  the  Table. 

This  table  gives,  to  the  radius  1,  the  values  of  the 
sine,  co-sine,  tangent,  and  co-tangent,  to  five  decimal 
places,  for  every  10'  from  0°  to  90°. 

For  sines  and  tangents,  the  degrees  are  given  in  the 
left  column,  and  the  minutes  at  the  top. 

For  co-sines  and  co-tangents,  the  degrees  are  given  in 
the  right-hand  column,  and  the  minutes  at  the  bottom. 
S.  N.  4. 


42  TRIGONOMETRY. 

51.    Problem. 

To  find  the  natural  sine,  co-sine,  tangent,  or  co-tangent 
of  a  given  arc  or  angle. 

Let  us  find  the  natural  sine  of  35°  42'  24". 

The  difference  between  the  natural  sines  of  35°  40' 
and  35°  50',  as  given  in  the  table,  is  .00236.  Now 


2'  24"  =  .24  of  10',  which  is  found  thus :     60 

10 


24 
2.4 


.24 

Then  take  Nat  sin  35°  40'=  .58307 

Correction  for  2'  24"  =  .00236  X  .24  =  .00057 

.-.     Nat  sin  35°  42'  24"=  .58364 

In  case  of  co-sine  or  co-tangent,  the  correction  must 
be  subtracted,  since,  between  0°  and  90°,  the  greater 
the  angle,  the  less  the  co-sine  and  co-tangent. 

52.   Examples. 

1.  Find  the  natural  sine  of  75°  45'  30". 

Ans.  .96927. 

2.  Find  the  natural  co-sine  of  15°  36'  12". 

Ans.  .96315. 

3.  Find  the  natural  tangent  of  43°  33'  18". 

Am.  .95079. 
'4.  Find  the  natural  co-tangent  of  84°  28'  30". 

Ans.  .09673. 

53.    Problem. 

To  find  the  angle  corresponding  to  a  given  natural  sine, 
co-sine,  tangent,  or  co-tangent. 

1.  Find  the  angle  corresponding  to  the  natural  sine 
.50754. 

Looking  in  the  table  we  find  the  angle  30°  30'. 


LOGARITHMIC  FUNCTIONS.  43 

2.  Find  the  angle  whose   natural   sine  —  .82468. 

The  next  less  sine,  sin  55°  30'  =  .82413. 

Difference  =         55 

Difference  corresponding  to  10'  =       164 

.  • .     Correction  =  10' X  7^  =  3'  21". 
164 

.  • .     Angle  =  55°  30'  +  3'  21"  =  55°  33'  21". 

In  case  of  co-sine  and  co-tangent,  the  angular  differ- 
ence must  be  subtracted,  since  the  greater  the  co-sine 
or  o«-tangent,  the  less  the  angle,  for  values  between  0° 
and  90°. 

54.   Examples. 

1.  Find  the  angle  whose  sine  is  .75684. 

Ans.  49°  11'  13". 

2.  Find  the  angle  whose  co-sine  is  .67898. 

Ans.  47°  14'  10". 

3.  Find  the  angle  whose  tangent  is  1.34567. 

Ans.  53°  22'  59". 

4.  Find  the  angle  whose  co-tangent  is  .98765. ' 

Ans.  45°  21'  22". 


TABLE  OF   LOGARITHMIC   FUNCTIONS. 

55.   Description  of  the  Table. 

The  table  of  logarithmic  functions  gives  to  the  radius 
10,000,000,000  the  logarithm  of  the  sine,  co-sine,  tangent, 
and  co-tangent,  for  every  minute,  from  0°  to  90°. 

The  expression,  logarithmic  sine,  tangent,  etc.,  is  equiv- 
alent to  the  logarithm  of  the  sine,  of  the  tangent,  etc. 

For  sines  and  tangents,  the  degrees  are  given  ut  the 
top  of  the  page,  and  the  minutes  in  the  left-hand 
column. 


44  TRIO  ONOMETR  Y. 

For  co-sines  and  co-tangents,  the  degrees  are  given  at 
the  bottom  of  the  page,  and  the  minutes  in  the  right- 
hand  column. 

The  columns  marked  D  1"  contain  the  difference 
for  1". 

50.   Problem. 

Find  the  logarithmic  sine  of  48°  25'  30". 

log  sin  48°  25'=  9.87390. 
D  1"  =  .19.     .  • .  Correc.  for  30"  =  .19  X  30  =  6 

.  •.  log  sin  48°  25'  30"  =9.87396 

In  case  of  co-sine  or  co-tangent,  the  correction  must 
be  subtracted,  since  between  0°  and  90°,  the  greater  the 
angle,  the  less  the  co-sine  and  co-tangent. 

57.   Examples. 

1.  Find  the  logarithmic  sine  of  75°  35'. 

Am.  9.98610. 

2.  Find  the  logarithmic  sine  of  25°  40'  24". 

Ans.  9.63673. 

3.  Find  the  logarithmic  co-sine  of  29°  55'  55". 

Ans.  9.93782. 

4.  Find  the  logarithmic  tangent  of  50°  50'  50". 

Ans.  10.08927. 

5.  Find  the  logarithmic  co-tangent  of  65°  45'  30". 

Am.  9.65349. 

58.   Problem. 

To  find  the  angle  corresponding  to  a  given  logarithmic 
sine,  co-sine,  tangent,  or  co-tangent. 


LOGARITHMIC  FUNCTIONS.  45 

Find   the   angle  whose   logarithmic  sine  =  9.84567 
For  next  less  we  have  sin  44°  3(X  =  9.84566 


D  1"  =  .21     .  • .  Correc.  =  1"  X        =  5",     .21)1.00(5. 
.•".     Angle  =  44°  30'  05". 

In  case  of  co-sine  and  co-tangent,  the  correction  for 
seconds  must  be  subtracted,  since  the  greater  the  co- 
sine or  co-tangent,  and  consequently  the  greater  the 
logarithm,  the  less  the  angle  for  values  between  0° 
and  90°. 

59.    Examples. 

1.  Find  the  angle  whose   logarithmic   sine    is  9.98437. 

Ans.  74°  43'  17". 

2.  Find  the  angle  whose  logarithmic  co-sine  is  9.78456. 

Ans.  52°  29'  19". 

3.  Find  the  angle  whose  logarith.  tangent  is   10.12346. 

Ans.  53°  02'  11". 

4.  Find  the  angle  whose  logarith.  co-tangent  is  9.99999. 

Ans.  45°  00'  03". 

<>0.    Problem. 

Given  any  natural  function,  to  find  the  corresponding 
logarithmic  function. 

1st  SOLUTION. 

Find  from  the  natural  function  the  corresponding 
angle;  then,  from,  the  angle,  the  corresponding  loga- 
rithmic function. 

2d  SOLUTION. 

Let  a  denote  any  arc  or  angle,  /(«)i  any  function 
of  a  to  the  radius  1,  and  /(a)*  the  corresponding 


4  6  TRIG  ONOMETR  Y. 

function   of  a   to   the   radius   R.     Then,  by  article   49 
we  have, 

/(a)*=/(o)i  X  R. 
Substituting  the  value  of  R   in  the  second  member, 

f(d)s=f(a)l  X  10,000,000,000. 
•••     log /(a)*  =  log /(a) i  +  10. 
Hence,  Add  10  ta  the  logarithm  of  the  natural  function. 

61.    Examples. 

1.  Given  nat.    sin    a  —  .98457,    required   a    and    iog 
sin  a.  Ans.  a  =  79°  55'  25",  log  sin  a  =  9.99325. 

2.  Given  nat.   cos    a  —  .63878,    required    a    and    log 
cos  a.  Ans.  a  =  50°  IT  52",  log  cos  a  =  9.80536. 

3.  Given  nat.    tan   a  =  1.68685,   required   a    and   log 
tan  a.  Ans.  a  =  59°  20  23",  log  tan  a  =  10.22708. 

4.  Given  nat.   cot   a  =  1.41987,    required  a   and   log 
cot  a.  Ans.  a  ==  35°  09'  24",  log  cot  a  =  10.15225. 

62.   Problem. 

Given  any  logarithmic  function,  to  find  the  corresponding 
natural  function. 

1st  SOLUTION. 

Find  from  the  logarithmic  function  the  correspond- 
ing angle ;  then,  from  the  angle,  the  •  corresponding 
natural  function. 

2d  SOLUTION. 
From  article  49  we  have, 

/wp^fp- 

.-.     \oef(a),  =  \osf(a)K—10. 


RIGHT  TRIANGLES. 


47 


Hence,  Subtract  10  from  the  logarithmic  function,  and  find 
the  number  corresponding  to  the  resulting  logarithm. 


63.   Examples. 

1.  Given   log  sin   a  —  9.87654,   required   a   and    nat 
sin  a,  Ans.  a  =48°  48'  44",  nat.  sin  a  =  .75255 

2.  Given   log    cos   a  =  9.84877,   required   a   and   nat 
cos  a.  Am.  a  =  45°  05'  41",  nat.  cos  a  =.70595 

3.  Given  log   tan  a  =  10.22708,   required  a  and  nat 
tan  a.  Ana.  a  =  59°  20'  23",  nat.  tan  a  =  1.68685 

4.  Given  log  cot  a  =  10.15225,   required  a  and  nat 
cot  a.  Ans.  a  =  35° 


'  24",  nat.  cot  a  =  1.41987. 


RIGHT   TRIANGLES. 


64.  Principles. 

PB   .    PK  :  :   HB   :   MK, 
or  h   :    1    :  :  p   :   sin  P. 
BP  :   £R   :  :   HP  :   SR, 

or  h  :   1   :  :   b   :   sin  B. 


.'.     (2) 


sn       = 


1.  Either  side  adjacent  to  the  right  angle  is  equal  to  the 
sine  of  the  opposite  angle  multiplied  by  the  hypotenuse. 

2.  The  sine  of  either  acute  angle  is  equal  to  the  opposite 
side  divided  by  the  hypotenuse. 


48  TRIGONOMETRY. 

Since  the  angles  P  and  B  are  complements  of  each 
other,  sin  P=cos  B,  and  sin  B  —  cos  P;  .  •.  (1)  and 
(3)  become, 


(3.)     4  ]-    and  (4) 


3.  Either  side  adjacent  to  the  right  angle  is  equal  to  the 
co-sine  of  the  adjacent  acute  angle  multiplied  by  the  hypot- 
enuse. 

4.  The  co-sine  of  either  acute  angle  is  equal  to  the  adja- 
cent side  divided  by  the  hypotenuse. 

PR   :    PN  :  :    HB    :    NL,  or  b    :    1    :  :   p    :    tan  P 
BH  :    BT  :  :    HP   :     TQ,  or  p  :    1   :  :    b    :    tan  B. 

tan  P  =  2-. 

(6Mt     n 

tan  B  =  — . 

5.  Either  side  adjacent   to   the  right  angle  is  equal  to  the 
tangent  of  the  opposite  angle  multiplied  by  the  other  side. 

6.  The  tangent  of  either  acute  angle  is  equal  to  the  oppo- 
site side  divided  by  the  adjacent  side. 

Since  the  angles  P  and  B  are  complements  of  each 
other,  tan  P=cot  B,  and  tan  B  =  cot  P;  .-.  (5)  and 
(6)  become, 

'cotB  =  £. 

(7)     ^  V    and  (8)     ' 


P 

7.  Either  side  adjacent  to  the  right  angle  is  equal  to  the 
co-tangent  of  the  adjacent  acute  angle  multiplied  by  the 
other  side. 


RI&HT  TRIANGLES.  49 

8.   The  co-tangent   of  either  acute  angle  is  equal  to  the 
adjacent  side  divided  by  the  opposite  side. 

BH   :    BT   :  :    BP   :    BQ,  or  p    :    1    : 
PH    :    PN    :  :    PB    :    PL,  or  b    :    1    : 


secP 


9.  Either  side  adjacent  to  the  right  angle  is  equal  to 
the  hypotenuse  divided  by  the  secant  of  the  adjacent  acute 
angle. 

10  The  secant  of  either  acute  angle  is  equal  to  the  hypot- 
enuse divided  by  the  adjacent  side. 

Since  the  angles  B  and  P  are  complements  of  each 
other  sec  B  =  cosec  P,  sec  P—  cosec  B;  .  '  .  (9)  and 
(10)  become, 

V  r» 

p  =  -  „>  cosec  P=  — 

P     I    and  (12)    J 


cosec      =  -J- 
cosec  B     J  b 

11.  Either  side  adjacent  to  the  right  angle  is  equal  to  the 
hypotenv.se   divided   by   the   co-secant  of   the   angle   opposite 
that  side. 

12.  The    co-secant  of  either  acute  angle   is    equal   to    the 
hypotenuse  divided  by  the  side  opposite  that  angle. 

Scholium.  By  some  authors,  principles  2,  4,  6,  8,  10, 
and  12,  have  been  given  in  the  form  of  definitions. 

Introducing  radius  into  these  formulas,  by  substitut- 
ing for  any  function  to  the  radius  1,  the  corresponding 
function  to  the  radius  R  divided  by  R,  and  reducing, 
we  have: 

S.  N.  5. 


50 


(1) 


TRIGONOMETRY. 


h  sin  P 


(2) 


sin     = 


sin  B  = 


(3) 


(4) 


cos     — 


D 

cos  P  =  -=— 
h 


(5) 


P  = 


R 


(6) 


Rb 

tan  E  = 


(7) 


b  cot  B 


(8) 


cot£  =  ^r- 
cot  P=  — 


(9) 


(10) 


sec      = 


cosec 


(12) 


cosec  P= 


cosec     =^ 


P 
Rh 


Applying  logarithms  to  these  formulas,  we  have: 

(1)     f    log    P  —  log    A  +  log    sin    P  —  10.    1 
I    log    b  —  log    h  4-  log    sin    £  —  10.    J 


f  log  sin    P=10-f  log  p  —  log     h.  1 

I  log  sin   B  =  10  +  log  b  —  log     h.  J 

f  log  p  =  log    h  -f  log  cos    5  —  10.  ) 

I  log  6  =  log    h  -f  log  cos    P  —  10.  j 


RIGHT  TRIANGLES.  51 


(^     (  log  cos  B  =  10  +  log  p  —  log    h.  } 

\  log  cos  P  =  10  -f  log  b  —  log     h.  ( 

^     (  log  jp  =  log    6  -f  log  tan    P  —  10.  1 

I  log  6  ==  log    p  +  log  tan    JB  —  10.  / 

(&)     (  log  tan  P=  10  -f  log  JD  -  log    6.  \ 

1  log  tan  B  =  10  +  log    b  —log    p.  J 

^     (  log  p  =  log    6  +  log  cot    5  —  10.  ) 

(log  b  =  log   p  +  log  cot    P  —  10.  J 

(8)     (  log  cot   5  =  10+  log    j>-log    b.  ) 

(  log  cot   P  =  10  +  log  b  —  log   p.  J 

/9)     f  log  ;>  =  =  10      +  log  h  —  log  sec  B.  ) 

1  log  6  —  10     +  log  h  —  log  sec  P.  j 

(10)     /  log  sec  B  ^  10  +  log  h  ~~  log    p'  \ 

\  log  sec  P  =  10  +  log  h  —  log    6.  J 


(11)    /    log    ^  ~  10      +  log  ^~~loS  cosec 

| 


log  •    6  =  10      +  log  h  —  log  cosec  B 

p.    1 

b.    f 


^og  cosec  P:=10+  log    h  --  log 
log  cosec  J5=  10  +  log    ^  --  log 


65.   Case  I. 

Given  the  •  hypotenuse  •  and  one  acute  angle,  required  the 
remaining  parts.  B 

(B. 

1.  Given  /^  =  365-         I    RequirJp. 
lP^33°12'.J 

5  =  90°  —  P=  90°  —  33°  12'  =  56°  48'. 

Either  side  adjacent  to  the  right  angle  is  equal  to  the  sine 
of  the  opposite  angle,  multiplied  by  the  hypotenuse. 

.  -  .        =  h  sin  P. 


52  TRIGONOMETRY. 

i      >  T-  h  sin  P 

Introducing  radius,  we  have,  p  —• ^~ 

A 

Applying  logarithms,  we  have, 

log  p  =  log  h  -f  log  sin  P  —  10. 

log  h  (365)  =  2.56229 

log  sin  P  (33°  12')  =  9.73843 
log  p  =  2.30072     .  • .    jt>  =  199.85. 

In   like   manner,  from  either  formula,   b  =  h   sin  B, 
or  b  =  h  cos  P,  we  find  b  =  305.41. 

rP^  40°  47' 40". 
ir.  I  b  = 


_ 
2   G  .      ^ 

f  5  =  62°  21' 10". 

3.  Given  {p^'^  }  RequiJ  ^=1018.512. 
IP^27    3850J  1^1944.364. 


66.   Case  II. 

Given  the  hypotenuse  and  one  side  adjacent  to  the  right 
angle,  required  the  remaining  parts.  « 


r  - 

1.  Given/  h  =  112'  I     Required^   A 
I  p.— *7.     J 


The  sine  of  either  acute   angle  is  equal  to  the  opposite 
side  divided  by  the  hypotenuse. 

.-.    sin  P=4' 
h 

Introducing  radius,  and  multiplying  by  J?,  we  have, 

r>        HP 

sin  P=--. 


EIGHT  TRIANGLES.  53 

Applying  logarithms,  we  have, 

log  sin  P  —  10  +  log  p  —  log  h. 

log  p  (97)  =1.98677 
log  h  (112)  =  2.04922 
log  sin  P  =9.93755  .-.  P=  60°  00'  17". 

B  =  90°  —  P=  90°  —  60°  00'  17"  **  29°  59'  43". 
b  =  h  sin  B,  or  b  =  h  cos  P,  .•'.  6  .=  55.991. 
We  can  also  find  b  as  follows: 


b  =  l/A2  — p2  =  1/(A  4-  p)  (h  —p). 
log  6  =  i  [  log  (A  +  P)  4-  log  (^  —  *>)]• 

(5  =  25°  47'  07". 

2.  Given  {  *  =r  7269'  1      Required  <{  P-  64^  12'  53". 

I  p  =  6545. 

rP-=19°  43'  36". 

3.  Given  /  h  ":  44^4  1      Required  4cB^  70°  16'  24". 

i*=H    j  I  6=  418.33. 


67.    Case  III. 

Given  one  side  adjacent  to  the  right  angle  and  one 
angle,  required   the  remaining  parts. 


=  152.67.  (B' 


U. 

90°  _  p=  90°  -  50°  18'  32"  =  39°  41'  28". 


54  TRIGONOMETRY. 

Either  side  adjacent  to  the  right  angle  is  equal  to  the  tan- 
gent of  the  opposite  angle  multiplied  by  the  other  side. 

.  • .    p  =  b  tan  P. 

Introducing   radius  and  applying  logarithms,  as   in 
the  preceding  cases,  we  find  p  =  183.95. 

Either  side  adjacent  to  the  right  angle  is  equal  to  the  co-sine 
of  the  adjacent  acute  angle  multiplied  by  the  hypotenuse. 

b 


b  =  h  cos  P;    .  • .     h  = 


cos  P 


Introducing  radius  and  applying  logarithms,  as  above, 
we  shall  find  h  =  239.05. 

2  Given  f  p==  3963'35  miles  =  the  earth's  radius. 

I  P=  57'  2.3"=  the  moon's  horizontal  parallax. 

Required  A,  the  distance  of  the  moon  from  the  earth. 

Ans.  h  =238889  miles. 

3  Q.-        f  P  —  3963.35  miles  =  the  earth's  radius. 

\  P=  8.9"  =  the  sun's  horizontal  parallax. 

Required  h,  the  distance  of  the  sun  from  the  earth. 

Ans.  h  =  91852000  miles. 

O  Q 

Scholium.     Sin  8.9"  =  sin    1'  X .  |jsr- 

.  •  -  log  sin  8.9"=  log  sin  I'-f  log  8.9  +  a.c.  log  60  —  10. 


68.   Case  IT. 

Given  the  two  sides  adjacent  to  the  right  angle,  required 
the  remaining  parts. 


OBLIQUE  TRIANGLES.  55 

The  tangent  of  either  acute  angle  is  equal  to  the  opposite 
#ide  divided  by  the  adjacent  side. 


Introducing  radius  and  applying  logarithms,  we  shall 
find  that  P=38°  13'  28". 

B  =  90°  —  P  =  90°  —  38°  13'  28"  =±  51°  46'  32". 

Either  side  adjacent  to  the  right  angle  is  equal  to  the  sine 
of  the  opposite  angle  multiplied  by  the  hypotenuse^ 

.  •  .    p  =  h  sin  P.         .  *  .     h=    .      ..  • 

sm  P 

Introducing  radius  and  applying  logarithms,  we  find 
h  =  47.466. 

(  P=4°  44'  37". 

2.  Given-/  *  =      ^73.  1     Required  <  5-85°  15'  23". 
to  =oo72il.  )  I  ,        OA^ 


I A  =8401 

3.  Given/  p"   J™'    }     Required  X  B=  45°  33'  43". 

(6   lUO.      J  \iiAA  nr* 


(P^47°39'0r. 

4.  Given  J  f:  V     Required^  5^42°  20'  53". 

I  6  =  15/5.  J  .       rt000, 


th=  144.253. 

I  B  =  42°  2(X 
^=2338.1. 


OBLIQUE   TRIANGLES. 

69.   Case  I. 

Given  one  side  and  two  angles,  required  the  remaining 
parts. 

Let   ABC  be   an  oblique  triangle, 
and  let  the  sides  opposite  the  angles 
A,  Bj  and  C  be  denoted  respectively    A 
by  a,  b  and  c. 


56  TRIGONOMETRY. 

Let  the  angles  A  and  B  and  the  side  a  be  given, 
and  the  angle  C  and  the  sides  b  and  c  be  required. 

We  find  C  from  the  formula, 

<7=180°—  (.4  +  5). 

Draw  the  perpendicular  p  from  the  vertex  C  to  the 
side  £,  thus  forming  two  right  triangles.  There  are 
two  cases: 

1st.  When  the  perpendicular  falls  on  the  side  c. 

From   the    principles  of  the  right 
triangle  we  have, 

p  =  b  sin  A  and  p  =  a  sin  B. 

.  ' .     b  sin  A  ==  a  sin  B. 

(1)     sin  A  :  sin  B  :  :  a  :  b. 

2d.  When  the  perpendicular  falls  on  c  produced. 
p  =  b  sin  A  and  p  —  a  sin  CBD. 

But  CBD  is   the    supplement   of 
CBA,  or  B  of   the   triangle.     Since  • 
the   sine   of   an   angle    is   equal   to 
the  sine  of  its  supplement, 


sin  CBD  =  sin  B;     .- .   p=±  a  sin  B. 

.'.     b  sin  A  =  a  sin  B. 

(1)  sin  A  :  sin  B  :  :  a  :  b. 

In  like  manner  we  may  find, 

(2)  sin  A  :  sin  C  :  :  a  :  c. 

Hence,  The  sine  of  the  angle  opposite  the  given  side  is  to 
the  sine  of  the  angle  opposite  the  required  side  as  the  given 
side  is  to  the  required  side. 

Introducing  radius  by  substituting  for  the  function 
to  the  radius  1,  the  corresponding  function  to  the 


OBLIQUE  TRIANGLES.  57 

radius  R  divided  by  J?,  and  reducing,  the  proportions 
(1)  and  (2)  will   be  of  the  same  form  as  before  substi- 
tution, and  hence  are  true  for  any  radius. 
From  proportions  (1)  and  (2),  we  find, 

*L     ,       a  sin  B  a  sin  C 

(3)     b  =  — r—  -r-  (4)     c  =  — ^—  -T-  •   • 

sin  ^4  sin  ^4 

Applying  logarithms  to  (3)  and  (4),  we  have, 

(5)  log  b  —  log  a  --f-  log  sin  B  -j-  a.  r.  log  sin  A  —  10. 

(6)  log  c  =  log  a  -f  log  sin  C  -\-  a.  c.  log  sin  A  —  10. 


70.    Examples. 

( ,4  =  35°  45'. 
1.  Given  <  B  =  45°  28'. 
I  a  =  7985. 

C=  ISO0  —  (A+  B)  =  18Q°  —  81°  13'  =  98°  47'. 

Since  the  sine  of  the  angle  opposite  the  given  side 
is  to  the  sine  of  the  angle  opposite  the  required  side 
as  the  given  side  is  to  the  required  side,  we  have  the 
proportion, 

.      _  j  r      a  sin  B 

sin  A  :  sin  B:\a\b,     . ' .  b  =  — : — 

sin  A 

.  • .     log  b  =  log  a  -\-  log  sin  B  -f  a.  c.  log  sin  A  —  10. 

log  a  (7985)  =  3.90227 

log  sin  B  (45°  28')  ==  9.85299 

a.c.  log  sin  A  (35°  45')  ==  0.23340 

log  b  -  3.98866     .  • .  b  =  9742.25. 

In  like  manner  we  have  the  proportion, 

a  sin  C 

sin  A  :  sin  C  :  :  a  :  c,     . ' .  c  =  -     — j-  • 

Sill    & 


58  TRIGONOMETRY. 

.'.    log  c  =  log  a  +  log  sin  C -f  «•  <*•  log  sin  A  —  10. 

log  a  (7985;  ==  3.90227 

log  sin  C  (98°  47')  =  9.99488 

a.  c.  log  sin  A  (35°  45')  =  0.23340 

log  c  =  4.13055    .  • .  c  =  13506.88. 

In  finding  log  sin  98°  47',  take  the  supplement  of 
98°  47,  which  is- 81°  13',  and  find  log  sin  81°  13'. 

f  A  =  50°  30'  40".  °\  c  C  =  58°  43'  50". 

2.  Given  <B  =  70°  45'  30".  V     Req.  <  b  =  585.2  yd. 

I  a  =  478.35  yd.  )  (  c  =  529.8  yd. 

f  B  =  65°  25'  35".  ^  (  A  =  54°  05'  51". 

3.  Given  I  C=  60°  28'  34".  V     Req.  <  c  =  11.72  miles. 

I  b  =  12.25  miles.  J  I  a  =  10.91  miles. 


71.  Case  II. 

Gicen  two  sides  and  an  angle  opposite  one  of  them,  re- 
quired the  remaining  parts. 

1.  WHEN  THE  GIVEN  ANGLE  is  ACUTE. 

Let  the  sides  a  and  6  and  the  angle  A  be  given,  and 
the  remaining  parts  be  required. 

Let  the  perpendicular  p  be 
drawn  from  C  to  the  opposite 
side.  Then  we  shall  have, 

p  =  b  sin  A. 

l.sl  If  a  >  p  and  a  <  b,  there  will  be  two  solutions. 

.For,  if  with  C  as  a  center  and  a  as  radius  a  circum- 
ference be  described,  it  will  intersect  the  side  opposite 
C  in  two  points,  B  and  &,  and  either  triangle,  ABC  or 
A B'C  will  fulfill  the  conditions  of  the  problem,  since 


OBLIQUE  TRIANGLES.  59 

it  will  have  two  sides  and  an  angle  opposite  one  of 
them  the  same  as  those  given.  Hence,  there  will  be 
two  solutions  if  a  has  any  value  between  the  limits 
p  and  b. 

2d.  If  a  =  p,  there  will  be  but  one 
solution. 

For,  as  a  diminishes  and  approaches    A- 
p,  the  two  points  B  and  B'  approach ; 
and  if  a  =  p,  B  and  B'  will  unite,  the  arc  will  be  tan- 
gent to  r,  and  the  two  triangles  will  become  one,  and 
there  will  be  one  solution. 

3d  If  a  —  6,  there  will  be  but  one 
solution. 

For,  as  a  increases  and  approaches 
/;,  the  points  B  and  B'  separate,  the 
triangle  ABC  increases,  and  the  triangle  AB'C  decreases; 
and  when  a  becomes  equal  to  b,  the  triangle  AB'C  van- 
ishes, and  there  remains  but  one  triangle,  or  there  is 
but  one  solution. 

4th.  If  a  >  b,  there  will  be  but  one 
solution. 

For,  although  there  are  two  tri- 
angles ABC  and  AB'C,  the  latter  is 
excluded  by  the  condition  that  the  given  angle  A  is 
acute,  since  CAB'  is  obtuse,  and  there  remains  but 
one  triangle  ABC  which  satisfies  the  conditions,  or 
there  is  but  one  solution. 

5th.  If    a   <  p,    there  will   be   no 
solution. 

For  the  arc  described  with  C  as 

center  and  a  as  radius  will  neither  intersect  the  oppo- 
site side  nor  be  tangent  to  it.  The  triangle  can  not 
be  constructed,  or  there  will  be  no  solution. 


60  TlilG  OXOMETR  Y. 


2.  WHEN  THE  GIVEN  ANGLE  is  OBTUSE. 

1st.  If  a  >  b  there  will  be  but  one  solution. 

For,  although  there  are  two  triangles  ABC  and  AFC, 
the  latter  is  excluded  by  the  condi- 
tions of  the  problem,  since  the  angle 
CAB'  is  acute  while  the  given  angle 
is  obtuse.  There  remains  but  one 
triangle,  ABC,  which  satisfies  all 
the  conditions  of  the  problem,  or  there  is  but  one  pos- 
sible solution. 

2d.  If  a .  =  b  there  will  be  no  solution. 

For  as  a  diminishes   and  approaches  6,   B  will   ap- 
proach A ;  and  when  a  becomes  equal 
to  6,  B  will   unite  with   J,  and  the 
triangle  ABC  will  vanish.     The  tri- 
angle AB'C  will  remain,  but  will  be 
excluded    by   the    conditions   of   the 
problem,  since  the  angle  CAB'  is  acute  while  the  given 
angle  is  obtuse. 

3(1.  If  a  <<  b  there  will  be  no  solution 

If  a  >  p  there  will  be  two  tri- 
angles, ARC  and  AB"C,  but  both 
are  excluded  by  the  condition  that 
the  given  angle  is  obtuse. 

If  a=p  the  two  triangles  reduce 
to  one,  right-angled  at  B,  which  is 
excluded  by  the  condition  that  the 
given  angle  is  obtuse. 

If  a  <<  p  no  triangle  can  be  con- 
structed with  the  given  parts,  and 
there  will  be  no  solution. 


OBLIQUE   TRIANGLES.  61 

72.   Summary  of  Results. 

1.  When  A  <  90°. 
Two  Solutions,     If  a  >  p  and  a  <  ft. 

r    Is*.  If  a=p. 

One  Solution,    -<    2rf.  If  a  =  b. 
I   3d.  If  a  >  6. 

No  Solution,     If  a  <  _p. 

2.  When  A  >  90°. 
One  Solution,     If  a  >  ft. 

/    1st.  If  a  =  ft. 
No  Solution,      |    ^  Ifa  <6> 

73.   Method  of  Computation. 

Reversing  the  order  of  the  couplets  of  the  proportion 
m  Case  I,  we  have 

(1)     a  :  ft  :  :  sin  A  :  sin  B. 

Hence,  The  side  opposite  the  given  angle  is  to  the  side 
opposite  the  required  angle,  as  the  sine  of  the  given  angle  is 
to  the  sine  of  the  required  angle. 

„      ft  sin  A 

(1)     gives         (2)     sin  B  = 

.-.  (3)     log  sin  B=  log  ft  -f  log  sin  A  -\-  a.  c.  log  a  —  10. 

If  there  is  but  one  solution,  take  from  the  table  the 
angle  B  corresponding  to  log  sin  B;  if  there  are  two 
solutions,  take  B  and  its  supplement  5',  for  both  cor- 
respond to  log  sin  B. 

We  find  C  from  the  formula, 

C—  180°  -.(A  +  B)  or  C=  180°  —  (A  -f  B'). 


62  TRIGONOMETRY. 

We  find  c  from  the  proportion, 

sin  A  :  sin  C  ::  a  :  r,     ^c  =  ^5-£- 

sm  J. 

.  • .     log  c  =  log  a  -j-  log  sin  C  -j-  a.  c.  log  sin  A  —  10. 

74.    Examples. 

(  a  =  9.25.       ^  (B. 

1.  GivJ   ft  =  12.56.      I  Req.<  C. 

I  ,4  =  30°  25'.  J  t  A 

p  =  b  sin  A 

Introducing  7?  and  applying  logarithms,  we  have 
log  p  =  log  6  -f-  log  sin  A  —  10. 

log  b  (12.56)  =  1.09899 

log  sin  A  (30°  25')  =  9.70439 
log  p  =  0.80338    .  • .  p  =  6.3589. 

Since  a  >  p  and  a  <  6,  there  are  two  solutions. 

Since  the  side  opposite  the  given  angle  is  to  the  side 
opposite  the  required  angle  as  the  sine  of  the  given 
angle  is  to  the  sine  of  the  required  angle,  we  have  the 
proportion, 

n  n       b  sin  A 

a  :  6  : :  sin  A  :  sin  B,     .  • .  sin  B  =  — 

a 

log  sin  B  =  log  b  -f  log  sin  A  -j-  a.  c.  log  a  —  10. 

log  b  (12.56)  =  1.09899 

log  sin  ^(30°  25')  =  9.70439 

a.  c.  log  a  (9.25)  =  9.03386 

log  sin  B  =9^3724     . ..    1  B  =  43'25'41". 

1  5'=  136°  34' 19". 


OBLIQUE  TRIANGLES. 
C==  180°  —  (A  +  B)  =  106°  9'  19", 
C'=  180°  —  (A  +  F)  =  13°  0'  41". 

a  sin 


63 


sin  A  :  sin  C 


sn 


log  c  =  log  a  -f  log  sin  C  -f-  a.  c.  log  sin  .4  —  10. 
Taking  the  value  of  (7,  we  have, 

log  a  (9.25)  ~?  0.96614 

log  sin  C  (106°  9'  19")  =9.98250 

o.c.  log  sin  A  (30°  25')        =0.29561 

log  c  =1.24425    .-.€  =  17.549. 

Taking  the  value  of  C",  we  have, 

log  a  (9.25)  =  0.96614 

log  sin  C'  (13°  0'  41")  =  9.35246 

a.  c.  log  sin  A  (30°  25')      =  0.29561 


log  c 


=  0.61421 


/-  a  =?  20.35. 
2.  Given  I  b  ==  20.35. 


Req. 


3.  Given 


A  =  52°  35'  27". 

a  =±  645.8. 
6  =  234.5. 
^  =  48°  35'. 


.  c  =  4.1135. 

'=52°  35'  27". 
'=74°  49' 06". 
==  24.725. 


^  15°  48'  04". 
Req      0=  115°  36'  56". 
I  c  =  776.53. 


|  o  =  17. 
4.  Given  <  6  =  40.25. 

I  ^.  =  27°  43'  15". 


r  a  =   94.26. 
5.  Given X   6  =  126.72. 


Req. 


Req. 


No  Solution. 
38°  52'  46". 

i4i°  r  14". 

c       f  113°  17' 14". 

\    11°    2'  46". 
_  f  185.439- 
"t   38.682. 


64  TRIG  OSOMETR  Y. 


\  (A  =  57°    O'.SO". 

6.  Given  -J  6  =  2000.         V     Req.  <  C  =  11°  44'  10". 
15  =  111°  15'. )  I  c  =  436.49. 

75.   Case  III. 

Given  tivo  sides  and  their  included  angle,  required  the 
remaining  parts.  Fj 

Let  ABC  be  a  triangle, 
and  let  the  sides  opposite 
the  angles  A,  B,  C,  be  de- 
noted, respectively,  by  a,  b, 
c.  Let  a  and  6,  and  their 
included  angle  C,  be  given,  and  the  remaining  parts, 
A,  B,  and  c,  required. 

The  sum  of  the  angles  A  and  B  is  found  from  the 

formula, 

A  +  B  =  180°  —  C. 

With  C  as  a  center,  and  b,  the  shorter  of  the  two 
given  sides,  as  a  radius,  describe  a  circumference  cut- 
ting a  in./),  a  produced  in  E,  and  c  in  H.  Draw  AE, 
AD,  CH,  and  DF  parallel  to  AE.  The  angle  DAE  is 
a  right  angle,  since  it  is  inscribed  in  a  semi-circle; 
hence,  its  alternate  angle,  ADF,  is  also  a  right  angle. 

The  angle  ACE  being  exterior  to  the  triangle  ABC, 
is  equal  to  A  -f  B.  But  ACE  having  its  vertex  at 
the  center,  is  measured  by  the  intercepted  arc  AE, 
The  inscribed  angle  ADE  is  measured  by  one-half 
the  arc  AE;  hence,  ADE  =  |  ACE  =  \  (A  -j-  B). 

CH  ==  CAj  since  they  are  radii  of  the  same  circle ; 
hence,  the  angle  CHA  =  A.  The  angle  CHA  being 
exterior  to  the  triangle  CHB  is  equal  to  HCB  -f-  B ; 
hence, 


OBLIQUE  TRIANGLES.  65 

But  HCB,  having  its  vertex  at  the  center,  is  meas- 
ured by  the  intercepted  arc  DH;  and  DAF,  being 
an  inscribed  angle,  is  measured  by  one-half  the  arc 
DH;  hence,  DAF  =  J  HCB  =  ±(A  —  B). 

In  the  right  triangles  ADE  and  ADF  we  have 

AE=AD  tan  ADE==--AD  tan  \(A  +  B). 
DF=AD  tan  DAF  =  AD  tan  $(A  —  B). 
From  the  similar  triangles,  ABE  and  F.RD,  we  have 
BE  :  BD  :  :  AE  :  DF. 

Since  <7£  =  CA,  BE  =  BC  +  CA  ==  a  -j-  6. 
Since  CD=CA,  BD=  BC—CA=a  —  b. 

Substituting  the  values  of  BE,  BD,  AE,  and  DF  in 
the  above  proportion,  and  omitting  the  common  factor 
AD  in  the  second  couplet,  we  have 

a-t-6  :  a  —  b  ::  tan  %(A  +  B)  :  tan  %(A  —  B). 


Hence,  In  any  plane  triangle,  the  sum  of  the  sides  in- 
chiding  an  angle  is  to  their  difference  as  the  tangent  of 
half  the  sum  of  the  other  two  angles  is  to  the  tangent  of 
half  their  difference. 

We  find  from  the  proportion,  the  equation 


.     log  tan  i  (A—B)  =  log  (a  —  b)  -f  log  tan 

-f  a.  r.  log  (a  -f  6)  —  10. 

We  have  now  found       A  +  B   and     (A  —  B. 


a  sin  C 

sin  A  :  sin  C  :  :  a  :  r,     .  • .  c  =  — : -r-  • 

sin  A 

.     log  c  =  log-a  -f-  log  sin  C-f-  a.  c.  log  sin  ^4  —  10. 

S.  N.  6. 


66  TRIGONOMETRY. 


76.   Examples. 


(a  =  37.56. 
1.  Given  <  b  =  23.75. 
lc=68°25' 


A+B  =  180°—  C=  111°  35'. 
a  +  6  :  a  —  b  ::  tan  £(^-|_£)  :  tan       ^_ 

.'.     tan       ^-B=(°"6)  te 


.'.     log  tan  i(v4—  B)  =  log  (a  —  6)  -f  log  tan 

+  .a,  c.  log  (a  +  b)  —  10. 

log  (a—  6)  (13.81)  :   1.14019 

log  tan  $(A+B)  (55°  47'  30")  =  10.16761 
a.  c.  log  (a+6)  (61.31)  =_8.21247 

log  tan  i  (^4—  B)  i   9^52027^ 

—5)^18°  19'  55". 


5)  =  74°    7'  25". 
B  =  ^  (4  +  5)  -  £  (4  -  B)  =  37°  27'  35". 

sin  ,4  :  sinC::  a  :  c,     .'.  c  =  a*inC. 

sin  A 

log  c  =  log  a  -f  log  sin  C  -f-  a.  f  .  log  sin  A  —  10. 

log  a  (37.56)  =  1.57473 

log  sin  C  (68°  25')  =  9.96843 

a.  c.  log  sin  A  (74°  7'  25")  =  0.01689 

log  c  1.56005,     .-.c=  36.312. 

f  a  =  996.63.         \  sA  =  66°  30'  37". 

2.  Given  1  b  =  712.83.  V    Req.  ^5  =  40°  59'  35". 

I  C=  72°  29'  48".  J  I  c  =  1036.35. 


OBLIQUE  TRIANGLES.  67 

<  b  =  776.525.  ^  f  B  =  115°  36'  56". 

3.  Given  1  c  =  234.5.  V  Req.  1  C  =  15°  48' 04". 

I  .4  =  48°  35'.  J  I  a  =  645.8. 

c  a  ==  11.7209.  \  fA  =  60°  25'  34". 

4.  Given  j  c  =  10.9232.  V  Req.  <  C  =  54°  08'  51". 

(  B  =  65°  25'  35".  J  1 6  ==  12.256.- 


77.  Case   IT. 

Given  the  three  sides  of  a  triangle,  required  the  angles. 

Let  ABC  be  a  triangle,  take  the 
longest  side  for  the  base,  and  draw 
the  perpendicular  p  from  the  vertex 
B  to  the  base. 

Denote  the  segments  of  the  base  by  s  and  s'  respect- 
ively. 

Then,'  (1)  c2—  s'2=p2,        and  (2)  a2—  s2=p2. 

.  • .  (6)  s  -f-  s'  :  a  -f-  c  : :  a  —  c  :  s  —  s'. 

Hence,  The  sum  of  the  segments  of  the  base  is  to  the  sum 
of  the  other  sides  as  the  difference  of  those  sides  is  to  the 
difference  of  the  segments. 

(6)8iyeB(7).W=.(a  +  ?.<a-C)- 


.-.  (8)  log  (s  —  «0  =  log  (a  +  c)  +  log  (a  — c) 

-j-  a.  c.  log  (s  -{-  s')  —  10. 

In  case  the  sides  of  the  triangle  are  small,  find  s  —  s' 
from  (7);  otherwise,  it  will  be  more  convenient  to  em- 
ploy (8). 


68 


TRIG  ONOMETR  Y. 


Having  s-\-sr  and  s  —  s',  we  find  s  and  s   thus, 

(9)  «  =  .£(«  +  *' 


(11)     cos  ^=  •-,         (12)     cos  C  =   -• 

c  C£ 

Introducing  #,  reducing,  and  applying  logarithms, 

(13)  log  cos  A  =--  10  -f  log  s'—  log  c. 

(14)  log  cos  C  —  10  -f-  log  s  —  log  a. 
From  which  we  find  A  and  C. 

Then,     (15)     B  =  180°  —  (4  +  C). 


1.  Given  <  b  =  150. 

U===ioa 


78.   Examples. 

.^|  r^4. 

Req.  <  5. 


a 


s  -}-  *'  :  a-j-c  ::  a  —  c  :  s  —  s'. 

,      (a  -f  c)  (a-c)      225  X  25        , 
^"" 


«  =  j  (.5  -f-  s')  -f  i  (*  —  s')  =  75  +  18.75  ^r  93.75. 
s'r=  I  (x  +  s')  —  i  (s  —  s')  =  75  —  18.75  =  56.25. 

cos  A  ~     '-  ,  or  introducing  7?,  cos  A  =  -- 
c  c 

.  •  .     log  cos  A  =  10  -j-  log  -s-'  —  log  c. 

log  s'  (56.25)  4=  1.75012 
log  <r  (100)  |=  2.QQOOQ 
log  cos  ^l  =9.75012  .;.  ^4  =  55°  46'  18". 

a  IDn 

cos  C=—  j  or  introducing  7?,  cos  (7=  —  • 
a  a 


HEIGHTS  AND  DISTANCES.  69 

.  • .     log  cos  C  =  10  -f  log  ,?  —  log  a. 

log  s  (93.75)  ==  1.97197 
log  a  (125)  =  2.09691 
log  cos  C  -9.87506  .-.  C=41°  24'  34". 

B  =  180  —  (A  +  C)  =  82°  49'  08". 

'  (  a  =  332.21.  ^  fA  =  66°  30'  35". 

2.  Given  <  b  =  345.46.  V      Required  1  B  =  72°  29'  53". 

I  c  =  237.61.  3  I C  =  40°  59'  32". 


ra  =  864. 
3.  Given  <  b  =^1308. 


f  a  =  251.25. 
4.  Given  <  6  =  302.5. 
(  c  =  342. 


rv4=,41°  00'  38". 

Required^  5  —  83°  25'  14". 

lc=55°  34'  08". 

(^^45°  22' 41". 

Required  ]  B  =  58°  58'  20". 

(  C  =  75°  38' 59". 


APPLICATION    TO    HEIGHTS    AND    DISTANCES. 
79.   Definitions. 

1.  A    horizontal    plane    is    a    plane    parallel    to    the 
horizon. 

2.  A  vertical    plane   is  a    plane    perpendicular   to   a 
horizontal  plane. 

3.  A  horizontal  line  is  a  line  parallel  to  a  horizontal 
plane. 

4.  A  vertical  line  is  a  line  perpendicular  to  a  hori- 
zontal plane. 

5.  A   horizontal   angle   is   an   angle  whose   plane  is 
horizontal. 


70 


TRIGONOMETRY. 


6.  A   vertical    angle    is    an    angle    whose    plane    is 
vertical. 

7.  An  angle  of  elevation  is 
a  verticle  angle,  one  of  whose 
sides    is    horizontal,    and    the 
inclined  side  above  the  hori- 
zontal side.     Thus,  BAG. 

8.  An  angle  of  depression  is  a  vertical  angle,  one  of 
whose  sides  is  horizontal,  and  the  inclined  side  below 
the  horizontal  side.     Thus,  DCA. 


80.   Problems. 

1.  Wishing   to  know  the   height  of  a  tree  standing 
on  a  horizontal  plane,  I  meas- 
ured from  the  tree   the   hori- 
zontal line   BA,   150  ft.,  and 

found  the  angle  of  elevation, 
BAC,  to  the   top   of   the  tree 
to  be  35°  20'.     Required  the 
height   of  the  tree. 
Ans.  106.335  ft. 

2.  In  surveying  a  tract  of  land,  I  found  it  impractic- 
able to  measure  the  side  AB 

on  account  of  thick  brush- 
wood lying  between  A  and  B. 
I  therefore  measured  AE,  7.50 
ch.,  and  EB,  8.70  ch.,  and 
found  the  angle  AEB  =  38°  46'.  Required  AB. 

Ans.  5.494  ch. 

3.  One  side  of  a  triangular  field  is  double   another, 
their   included  angle   is   60°,  and  the  third  side  is  15 

ch.     Required  the  longest  side. 

Am.  17.32  ch. 


HEIGHTS  AND  DISTANCES. 


71 


4.  Wishing  to  know  the  width  of  a  river,  from  the 
point  A  on   one  bank  to  the 

point  C  on  the  other  bank, 
I  measure  the  distance  AB, 
75  yd.,  and  find  the  angle 
BAC  =  87°  28'  30",  and  the 
angle  ABC  =  47°  38'  25". 
Required  AC,  the  width  of 
the  river.  Ans.  78.53  yd. 

5.  I  find  the  angle  of  elevation,  BA C,  from  the  foot 
of  a  hill  to  the  top  to  be  46°  25'  30".    Measuring  back 
from   the   hill,    AD  =  500   ft., 

I  find  the  angle  of  elevation 
ADC  =25°  38'  40''.  Required 
BC,  the  vertical  height  of  the 
hill.  Ans.  441.87  ft. 

6.  From  .the  foot  of  a  tower  standing  at  the  top  of  a 
declivity,  I   measured  AB 

=  45  ft.,  and  the  angle 
ABD  =  50°  15'.  I  also 
measured,  in  a  straight 
line  with  AB,  BC=68  ft., 
and  the  angle  BCD  =  30° 
45'.  Required  AD,  the 
height  of  the  tower.  Ans.  82.94  ft. 

7.  Wishing  to  know  the  height  of  a  tower  standing 
on  a  hill,  I  find  the 

angle  of  elevation, 
BA  C,  to  the  top  of 
the  hill  to  be  44°  35', 
and  the  angle  of  ele- 
vation to  the  top  of 
the  tower  to  be  59° 
48'.  Measuring  the 
horizontal  line  AE,  275  ft.,  I  find  the  angle  of  eleva- 


72 


TRIGONOMETRY. 


tion  to  the  top  of  the  tower  to  be  46°  25'.     Required 
the  height  of  the  tower.  Ans.  317.143  ft. 


DC     =  24  ch. 
CDB  =  45°. 


8.  Given 


Required 
9.  Given 


DCA  =48°. 
ACB  =60°. 

=  38.61  ch. 


=  800  yd.,  4C=600  yd 
ADC=ZZ°    45',    BDC =22°   30'.    Re- 
quired DA,  DC,  DB. 

Ans.  £4=710.15  yd.,  DC=  1042.5 
yd.,  DB  =  934.28  yd. 

Remark. — Describing  the  circumfer- 
ence through  4,  B,  D,  and  drawing 
AE  and  BE,  EAB  =  BDC,  EBA  =  ADC. 


RELATIONS  OF  CIRCULAR  FUNCTIONS. 
81.    Fundamental  Formulas. 

Let  a  =  the  angle  OCT  =  the  arc  OT,  and  CO  = 


=  1.  Then,  we  have  MT  =  CN=  sin 
a,  AT  —  C3f  =  cos  a,  MO  =  vers  a, 
JVO'  =  covers  a,  OR  =  tan  a,  O'S'  = 
cot  a,  CR  =  sec  a,  CS=cosec  a. 

By  articles  39—46,  sin  (90°  —  a)  = 
cos  a,  cos  (90°  —  a)  =  sin  a,  etc. 

From  the  diagram  we  have 


o' 


+  CM   =CT 

Substituting  the  values  of  MT,  CM,  and  CT,  we  have 
(1)     sin2o  +cos2a  =  l. 


CIRCULAR   FUNCTIONS.  73 

Hence,  The  square  of  the  sine  of  any  arc  plus  the  square 
of  its  co-sine  is  equal  to  1. 

From  (1)  we  have,  by  transposition, 

(2)  sin2a=  1  —  cos2  a, 

(3)  cos2  a  =  1— sin2  a.     Hence, 

1.  The  square  of  the  sine  of  any  arc  is  equal  to  1  minus 
the  square  of  its  co-sine. 

2.  The  square  of  the  co-sine  of  any  arc  is  equal  to  1  minus 
the  square  of  its  sine. 

From  the  diagram  we  have 

MO  =  CO—  CM. 
Substituting  the  values  of  MO,  CO,  and  CM,  we  have 

(4)  vers  a  —  1  —  cos  a. 

Hence,   The  versed-sine  of  any  arc  is  equal   to  1  minus 
its  co-sine. 

.  • .     vers  (90°  —  a)  =  1  —  cos  (90°  —  a). 
.  • .     (5)     covers  a  —  1  —  sin  a. 

Hence,  The  co-versed-sine  of  any  arc  is  equal  to  1  minvx 
its  sine. 

From  the.  diagram  we  have 

CM     :  CO  ::  MT      :  OR, 

or  cos  a  :     1     :  :  sin  a  :  tan  a. 

,«,  sin  a 

.' .     (6)     tan  a  = 

cos  a 

Hence,    The   tangent    of  any  arc    is    equal  to   its  sine 
divided  by  its  co-sine. 

S.  N.  7. 


74  TRIGONOMETRY. 


cos  a 

.  •  .     (7)     cot  a  =  -r  --- 
sin  a 

Hence,  The  co-tangent  of  any  arc  is  equal  to  its  co-sine 
divided  by  its  sine. 

(6)  X  (7)  =  (8)     tan  a  cot  a  =  1. 

Hence,  The  tangent   of  any  arc  *  into  its   co-tangent   is 
equal  to  1. 

1 


(8)  -f-  cot  a  =  (9)     tan  a  = 


cot  a 


Hence,  The  tangent  of  any  arc  is  equal  to  the  reciprocal 
of  its  co-tangent. 

1 


(8)  -^  tan  a  =  (10)      cot  a  = 


tan  a 


Hence,  The  co-tangent  of  any  arc  is  equal  to  the  recip- 
rocal of  its  tangent. 

CM  :  CO   ::  CT  :  CR,   or  cos  a  :  1  : :  1  :  sec  a. 

1 


.  • .     (11)     sec  a  — 


cos  a 


Hence,  The  secant  of  any  arc  is  equal  to  the  reciprocal 
of  its  co-sine. 

.'.    sec  (90°  —  a)  = 


.  *  .     (12)   cosec  a  — 


cos  (90°—  a) 


sin  a 


Hence,  The  co-secant  of  any  arc  is  equal  to  the  recip- 
rocal of  its  sine. 


.'.     (13)     sec2a  =  l  -f  tan2  a. 


CIRCULAR  FUNCTIONS. 


75 


Hence,  The  square  of  the  secant  of  any  arc  is  equal  to  1, 
plus  the  square  of  its  tangent. 

.-.     sec2  (90°—  a)  ==  1  +  tan5  (90°— a). 
.  • .     (14)     cosec2  a  =  l  -\-  cot2  a. 

Hence,  The  square  of  the  co-secant  is  equal  to  1,  plus  the 
square  of  the  co-tangent. 


82.  Summary  of  Fundamental  Formulas. 


1. 

sin2 

a 

+  COS 

2/7                1 

9. 

1 

cot  a 

2. 

sin2 

a, 

—  \  —  cos2  a. 

10 

1 

3. 

cos2 

a 

=  1  —  sin2  a. 

tan  a 

4. 

vers 

a 

-cos  a.. 

11. 

1 

cos  a 

5 

covers 

a  —  1 

—  sin  a. 

19 

1 

LL. 

cosec  a  - 

~  sin  a 

p. 

, 

sin 

a 

. 

cos 

a 

7. 

cot  a  - 

cos 

a 

13. 

sec2  a  — 

1  -+-  tan2  a. 

sin 

a 

8. 

tan 

a 

cot  a  =  l. 

14. 

cosec2  a 

=  1  -h  cot2  a. 

83.   Problems. 

1.  Prove  that  the  above  formulas  become  homogene- 
ous by  the  introduction  of  R. 

2.  Deduce   formulas  (5),  (7),  (12)  and  (14)  from  the 
diagram. 

3.  Prove  that  the  above  formulas  are  true  if  a  is  in 
the  second,  third?  or  fourth  quadrant. 


76 


TRIGONOMETRY. 


84.   Each  Function  in  Terms  of  the  Others. 


sin 
sin 
sin 

sin 
sin 

sin 

sin 

cos 

cos 
cos 

cos 
cos 

COS 

COS 

""] 

a  —  1/1  —  cos2  a. 

vers  a=l—V  1—  sin2  a. 

. 

vers  a=l—  cos  a. 

a  =1/2  vers  a  —  vers2  a. 

a  —  1  —  covers  a. 
tan  a 

vers  a  =  1  —  l/  2  cvs  a  —  cvs2  a. 
vers  a  —  1  

1/1+  tan2  a 
a  — 

1/1+  -tan2  a 
vers  a  -  1            Cot  a 

VI  -f  cot2a 

V  14-  cot2  a 

sec  a  —  1 
vers  a  —  — 

V7sec2a  —  1 

sec  a 
1 

sec  a 

V  cosec2  a  —  1 

VPTS  (1  1                                                 • 

cosec  a 

cosec  a 
covers  a  =  l  —  sin  a. 

a  =1/1  —  sin2  a. 
a  =  1  —  vers  a. 

covers  a=l  —  1/1  —  cos2  a. 

a  =1/2  cvs  a  —  cvs2  a. 
1 

cvs  a  =  l  —  1/2  vs  a  —  vs2  a. 
tan  a 

l/l  -f  tan2  a 
cot  a 

d  

1/1+  tan2  a 
onvpi"^  <7  —  1  —  • 

1/1-j-  cot2  a 
1 

1/1-f-COt2  ft 

V  sec2  a—  1 

sec  a 

sec  a 
cosec  a  —  1 

pnvpv^:  /7  —   -  •            -..»-    i 

I/  cosec2  a  —  1 

cosec  a 

cosec  a 

CIRCULAR  FUNCTIONS. 


77 


84.    Each  Function  in  Terms  of  the  Others. 


tan 

tan 
tan 

tan 

tan 
tan 
tan 

cot 
cot 

cot 

cot 
cot 

cot 

1 

cot 

sin  a 

SPP   n  —    ••• 

1/1  —  sin2  a 

1/1  —  sin2  a 
sec  ft  —  

V  1—  cos2  a 

a=  • 
cos  ft 

COS  ft 

Sjpp     /7    . 

1/2  vs  a  —  vs-  a 

— 

1  —  vers  a 

Sf*P    r/  _  —  ^— 

1  —  vs  a 
1  —  cvs  a 

V  2  cvs  ft  —  cvs2  a 
1 

sec  ft—  1/1  +  tan2  ft. 

cot  ft 

sen  ,^l/l+cot2ft 

ft  ;=  I/  sec2  ft  —  1. 
1 

cot  ft 
cosec  a 

V  cosec  ^  a  —  1 

I7  cosec2  a  —  1 

1 

cosec  ft      —  :  • 

1/1  —  sin2  a 

sin  a 

COS   ft 

sin  a 

1 

POCJPP  (7  —           •••-  •  - 

1/1  —  cos2  ft 

IXTO    /]• 

1/1  —  cos2  ft 

1 

1/2  vs  ft  —  vs2  « 

1/2  vs  ft  —  vs2  a 
1 

1  —  CVS  ft 

1 

1/1  +  tan2  a 

ft  —  , 
tan  ft 

1 

tan  ft 

cosec  a  =  -|/l-f-  cot2  ft. 
sec  ft 

I/  sec2  «  —  1 

ft  =  1/cosec2  ft  —  1. 

I/  sec2  ft  —  1 

78 


TRIGONOMETRY. 


85.    Functions  of  Negative  Arcs. 


We  first  find  the  sine  and  co-sine 
of  —  a,  in  terms  of  the  functions  of  a 
from  the  diagram.  Then,  dividing  the 
sine  by  the  co-sine,  the  cosine  by  the 
sine,  taking  the  reciprocal  of  the  co- 
sine and  the  reciprocal  of  the  sine, 
we  have 


sin  ( —  a)  = 
tan  ( —  a)  = 
sec  ( — a)  = 


—  sin  a, 


cos     ( —  a)  = 


cos  a, 

tan  a,        cot     ( —  a)  —  —  cot  a, 
sec  a,        cosec  ( —  a)  —  —  cosec  a. 


86.    Functions  of  (n  90°  +  a). 

1.  Let  n  be  1  and  a  be  negative. 

From   the   figure  of  the  last  article,  and   by  similar 
processes, 

sin  (90° — a)=cos  a,  cos  (90°  —  a)  =  sin  a, 
tan  (90°  —  a)  =  cot  a,  cot  (90°  —  a)  =  tan  a, 
sec  (90°  —  a)  =  cosec  a,  cosec  (90°  —  a)  =  sec  a. 

These    relations    have    already  been    found,    articles 
39—46. 


2.  Let  n  be  I  and  a  be  positive. 


sin  (90°  -f-  a)  =  cos  a, 
tan  (90°  +  a)  =  — cot  a, 
sec  (90°  -f  a)  =  —  cosec  a, 


cos  (90°  -f  a) 
cot  (90°  -f  a) 
cosec  (90°  +  a) 


3.  Let  n  be  2,  and  a  be  negative. 

sin  (180°—  a)  ==  sin  a,  cos  (180°— a)  = 
tan  (180°—  a)  =  —  tan  a,  cot  (180°—  a)  = 
sec  (180°—  a)  .=  —  sec  a,  cosec  (180°—  a)  = 


—  sin  o, 

—  tan  o, 
sec  a. 


cos  a, 
cot  a, 
cosec  a. 


CIRCULAR  FUNCTIONS.  79 

4.  Let  n  be  2,  and  a  be  positive. 

sin  (180°  -f  a)  =  -  sin  a,  cos  (180°  -f-  a)  =  -  cos  a, 
tan  (180°+  a)  =  tan  a,  cot  (180°-f-  a)  ==i  cot  a, 
sec  (180°  -f  a)  =  —  sec  a,  cosec  (180°  -j-  a)  —  —  cosec  a. 


5.  Let  ?i  be  3,  and  a  be  negative. 

sin  (270°—  a)  =  -  cos  a,  cos  (270°—  a)  =  —  sin  a, 
tan  (270°—  a)  ==  cot  a,  cot  (270°—  a)  sM  tan  a, 
sec  (270° —  a)  —  —  cosec  a,  cosec  (270°—  a)  =  -  sec  a. 

6.  Let  91  be  3,  and  a  be  positive. 

sin  (270°  -f  a)  =  -  cos  a,  cos  (270°+  a)  =  sin  a, 
tan  (270°+  a)  =  -  cot  a,  cot  (270° -f-  a)  =  —  tan  a, 
sec  (270°  -f-  «)  —  cosec  a,  cosec  (270° -(-  a)  =  —  sec  a. 

7.  Let  w  be  4,  and  a  be  negative. 

sin  (360°—  a).  =  —  sin  a,  cos  (360°—  a)  =  cos  a, 
tan  (360°—  a)  =  —  tan  a,  cot  (360°—  a)  =  —  cot  a, 
sec  (360°—  a)  —  sec  a,  cosec  (360°  —  a)  =  —  cosec  a. 

8.  Let  n  be  4,  and  a  be  positive. 

sin  (360°+  a)  =  sin  a,  cos  (360° -f  a)  ^  cos  a, 
tan  (360°+  a)  jes  tan  a,  cot  (360°+  a)  ±5  cot  a, 
sec  (360° -f  a)  =  sec  a,  cosec  (360°+  a)  ==  cosec  a. 

It  will  be  observed  that  when  n  is  even,  the  func- 
tions in  the  two  members  of  the  equations  have  the 
same  name;  and  that  when  n  is  odd,  they  have  con- 
trary names.  The  algebraic  sign  attributed  to  the  sec- 
ond member  is  determined  by  the  quadrant  in  which 
the  arc  is  situated. 

Let  this  article  be  reviewed,  and  these  principles 
applied  in  determining  the  names  and  algebraic  signs 
of  the  second  members. 


80  TRIGONOMETRY. 

Hence,  functions  of  arcs  greater  than  90°  can  be  found 
in  terms  of  functions  of  arcs  less  than  90°.     Thus, 

1.  sin  120°  =  sin  (  90° -f  30°)  =      cos  30°. 

2.  cos  290°  =  cos  (270°  +  20°)  =       sin  20°. 

3.  tan  165°  =  tan  (180°  —  15°)  =  —  tan  15°. 

If  n  is  integral  and  positive,  prove  the  following: 

4.  sin  In  180°  +  (—  1)"  a]  =  sin  a. 

5.  cos  (n  360°  ±  a)  ==  cos  a. 

6.  tan  (n  180°  +  a)  =  tan  a. 

7.  Any  function  of  (n  360°  -f  «)  —  the  same  func- 
tion of  a,  whatever  be  the  value  of  a. 


87.  Values  of  Functions  of  Particular  Arcs. 

1.   To  find  the  functions  of  30°. 

Since  60°  is  one-sixth  of  the  circumference,  the  chord 
of  60°  is  equal  to  one  side  of  a  regular  inscribed  hex- 
agon, which  is  equal  to  the  radius  or  1.  But  the  sine 
of  30°  is  equal  to  one-half  the  chord  of  60°. 


.-.    (1)    sin  30°=  |,     ./.    (2)   cos30°=:l/l— ^=£1/3. 

Dividing   (1)   by  (2),   then    (2)    by   (1),    taking   the 
reciprocals  of  (2)  and  (1),  we  have 

(3)     tan  30°  =  — L ,       (4)     cot  30°  ==  l/^ 
1'    o 

(5)     sec  30°  =  -JL ,       (6)     cosec  30°  =  2. 
V   o 

2.    To  find  the  functions  of  60°. 

From  article  40,  sin  60°  =  sin  (90°  —  30°)  =  cos  30°, 
cos  60°  =  cos  (90°  —  30°)  =  sin  30°.     Hence, 


CIRCULAR  FUNCTIONS.  81 

(1)     sin  60°  =  il/~3;        (2)     cos      60'=$, 
(3)     tan  60°  =  i/~3",  (4)     cot      6( 


(5)    sec  60°  =  2, 


V73 

(6)    cosec  60°  =  -^L . 


3.   To  find  the  functions  of  45°. 

From   Art.  40,  sin  45°  =  sin  (90°  —  45°)  =  cos  45°  ; 
but  sin2  45°  +  cos2  45°  ==  1', 


2  sin2  45°  =1, 

(1)     sin  45°  =  -}  1/27 

(3)     tan  45°  =  1, 

(5)     sec  45°  =  1/27 


sin2  45°  =  f     Hence, 
(2)    cos  45°  ==  il/2T 
(4)     cot  45°  =  1, 
(6)     cosec  45°  =  V  27 


5.    cosec  210°  ±±  —  2. 


240°  =  — = . 
V  3 


Prove  the  following : 

1.  sec  120°  =  —  2. 

2.  cos  135°=— £  1/27 

3.  sin  300°^   -^1/37 

4.  tan  225°=        1. 

9.  Construct  an  angle  whose  tangent  is  —  1. 

10.  Construct  an  angle  whose  sine  is  —  \. 

11.  Find  all  the  functions  of  150°. 


6.  cot 

7.  sin     390°  =  f 

8.  cos(— 1203)=— f 


88.    Inverse  Trigonometric  Functions. 

If  x  —  sin  a,  then  a  is  the  angle  or  arc  whose  sine 
is  x,  which  is  written  a  ==  sin"1  x,  and  read  a  equals 
the  arc  whose  sine  is  x. 


82  TRIG  ONOMETR  Y. 

It  must  not  be  supposed  that  ~]  is  an  exponent,  and 

that  sin"1  x  = ;  this  would  be  a  grievous  error. 

sm  x 

Let  the  following  be  read : 
cos"1^,  tan"1.?,  sec"1:*;,  cosec"1^,  sin-1(cosa:),  sin(sin~'j}, 

sin^z^cosec"1 — ,  cos"1  x  =  sec"1 — ,  tan"1  x  =  cot"1—. 
x  x  x 

The  above  notation  is  not  altogether  arbitrary;  for 
let  f(x)  be  any  function  of  x,  and  let  /[/(x)],  or,  nioro 
briefly,  let  f2(x)  be  the  same  function  of  /(a?),  which 
notation  denotes,  not  the  square  of  /(#)>  that  is,  not 
[/(a?)]2,  but  that  the  same  function  is  taken  of  f(x)  as 
of  x.  Thus,  if  f(x)  =  sin  x,  /[/(&)]  =  sin  (sin  x), 
then,  in  general, 

(1)  /"•/»  (*)=/"*;•  0). 

If  n  =0,  (1)  becomes, 

(2)  /- /°  (*)=/"•(*)• 
.-.     (3)    /'(*)  =  *. 

If  m  —  1,  and  n  =     -  1,  (!)•  becomes, 
(4)    ff-l(x)=f°(x)=x. 

Hence,  f~l(x)  denotes  a  quantity  whose  like  func- 
tion is  x. ' 

Hence,  if  y=^sin~lx,  sin  y  =  sin  (sin'}x)=x;  that 
is,  y  or  sin-1x  is  an  arc  whose  sine  is  x. 

It  would  follow  from  the  above  that  sin2  a  ought  to 
signify  sin  (sin  a),  and  not  (sin  a)2;  but  since  we 
rarely  have  sin  (sin  a),  it  is  customary  to  write  sin2  a 
for  (sin  a)2,  as  we  are  thus  saved  the  "trouble  of  writing 
the  parenthesis. 


CIRCULAR  FUNCTIONS. 


Ifc  would  not,  of  course,  do  to  write  sin  a2  for  (sin  a)2, 
for  then  we  should  have  the  sine  of  the  square  of  an 
arc  for  the  square  of  the  sine  of  an  arc. 


Let  the  following  equations  be  proved : 


1. 


2.     sin~1^  =  |tan~1V/3. 


4.     cos-1i  =  2cot-1V/3. 
5..    sin-1 1  =  2  tan- >1. 


89.   Problem. 

To  find  the  sine  and  co-sine  of  the  sum  of  two  angles. 

Let   a  =  the   angle  OCA,  and  b  =  the   angle    ACS. 
Draw  BL  perpendicular  to  CA,  BP  and 
LM  perpendicular  to  CO,  and  LN  parallel 
to  CO. 

The  triangles  NBL  and  MCL  are  sim- 
ilar, since  their  sides  are  respectively 
perpendicular;  hence,  the  angle  NBL  opposite  the  side 
NL  equals  the  angle  MCL  opposite  the  homologous 
side  ML.  But  MCL  =  a;  hence  NBL  =  a, 

From  the  diagram  we  find  the  following  relations: 

(1)  LB  =  sin  6. 

(2)  CL  =  cos  6. 

(3)  PB  =  ML  +  NB. 

(4)  PB  =  sin  OCB  ±±  sin  (a  -f  6). 

(5)  ML  =  sin  MCL  X  CL  =  sin  a  cos  b. 

(6)  AT£  =35  cos  NBL  X  LB  =  cos  a  sin  b. 

Substituting  the  values  of  PB,    ML,  and  NB,  found 

in  (4),  (5),  and  (6),  in  (3),  and  denoting  the   formula 
by  (a),  we  have 

(a)     sin  (a  -f  6)  =  sin  a  cos  b  +  cos  a  sin  6. 


84  TRIGONOMETRY. 

Hence,  The  sine  of  the  sum  of  two  angles  is  equal  to  the 
sine  of  the  first  into  the  co-sine  of  the  second,  plus  the  co- 
sine of  the  first  into  the  sine  of  the  second. 

From  the  diagram  we  find  the  follow- 
ing relations: 

(1)     CP  =  CM  —  NL. 


(2)  CP  =  cos  OCB  =  cos  (a  +  6). 

(3)  CM=  cos  MCL  X  CL  =  cos  a  cos  b. 

(4)  NL  =  sin  NBL  X  LB  =  sin  a  sin  b. 

Substituting   the   values  of  CP,    CM,  and  NL,  found 
in  (2),  (3),  and  (4),  in  (1),  we   have 

(6)     cos  (a  -j-  6)  =  cos  a  cos  b  —  sin  a  sin  b. 

Hence,  The  co-sine  of  the  sum  of  two  angles  is  equal  to  the 
product  of  their  co-sines  minus  the  product  of  their  sines. 


90.    Problems. 

1.  Prove  that  formulas  (a)  and  (b)  become  homogene- 
ous by  introducing  R. 

2.  Prove    that    formulas  (a)  and   (b)   are    true    when 
(a  -f  ft)  is  in  the  second  quadrant, 

3.  Prove    that    formulas   (a)    and  (6)   are    true   when 
(a  -j-  6)  is  in  the  third  quadrant. 

4.  Prove    that    formulas    (a)   and  (b)  are    true    when 
(a  -|-  6)  is  in  the  fourth  quadrant. 

5.  Deduce   formula  (6)  from   formula  (a)  by  substitu- 
ting  90°  —  a  for   a,  .and   — 6   for   b,   and    reducing   by 
articles  85—86. 

6.  Develop  sin  (45°-^  30°)  by  formula  (a). 

i 

7.  Develop  cos   105°  by  formula  (b). 


CIRCULAR  FUNCTIONS.  85 

91.   Problem. 

To  find  the  sine  and  co-sine  of  the  difference  of  two  angles. 

Let  a  —  the  angle  OCA,  and  6  =  the  angle  EC  A. 

Draw  BL  perpendicular  to  CA,  LP 
and  BM  perpendicular  to  CO,  and  BN 
parallel  to  CO. 

The  triangles  NLB  and  PCX  are  sim-    c  __ 
ilar,   since   their    sides   are    respectively 
perpendicular;   hence,  the  angle  NLB,  opposite  the  side 
NB,   equals  the    angle    PCL   opposite    the    homologous 
side   PL.     But   the   angle   PCL  =  a ;    hence,   the    angle 
NLB  =  a.     Then  we  shall  have 

(1)  LB  =  sin  ft. 

(2)  CL  =  cos  b. 

(3)  MB  =  PL  —  NL. 

(4)  MB  F=  sin  OCB  =  sin  (a  —  6). 

(5)  PL  --=  sin  PCYL  X  CL  =  sin  «  cos  b. 

(6)  JVL  =:  cos  NLB  X  LB  =  cos  a  sin  6. 

Substituting  the  values  of  MB,  PL,  and  NL,  found  in 

(4),  (5),  and  (6),  in  (3),  we  have 

(c)     sin  (a.  —  6)  —  sin  a  cos  6  —  cos  «  sin  b. 

Hence,  The  sine  of  the  difference  of  two  angles  is  equal 
to  the  sine  of  the  first  into  the  co-sine  of  the  second,  minus 
the  co-sine  of  the  first  into  the  sine  of  the  second. 

From  the  diagram  we  find  the  following  relations: 

(1)  CM=CP+NB. 

(2)  CM  =  cos  OCB  =  cos  (a  —  6). 

(3)  CP  =  cos  PCL  XCL=--  cos  a  cos  b. 

(4)  NB  =  sin  NLB  X  LB  =  sin  a  sin  b. 


86  TRIGONOMETRY. 

Substituting   in  (1)  the  values  of  CM,  CP,  and  NB 
found  in  (2),  (3),  and  (4),  we  have 

(d)     cos  (a  —  6)  =  cos  a  cos  6  +  sin  a  sin  b. 

Hence,  The  co-sine  of  the  difference  of  two  angles  is  equal 
to  the  product  of  their  co-sines,  plus  the  product  of  their  sines. 


92.  Problems. 

1.  Prove  that  formulas  (r)  and  (d)  become  homogene- 
ous by  introducing  R. 

2.  Deduce  formulas  (c)  and  (d)  from  (a)  and  (6),  re- 
spectively, by  substituting  —  b  for  b,  and  reducing  by 
article  85. 

3.  Prove   that   formulas   (c)   and  (d)  are   true   when 
(a  —  6)  is  in  the  second  quadrant. 

4.  Prove    that    formulas   (c}   and  (d)  are   true   when 
(a  —  6)  is  in  the  third  quadrant. 

5.  Prove    that    formulas    (r)  and   (d)  are   true   when 
(a  —  b)  is  in  the  fourth  quadrant. 

93.  Problem. 

To  find  the  tangent  and  co-tangent  of  the  sum  or  differ- 
ence of  two  angles. 

Dividing  (a)  by  (6),  we  have 

sin  (a  -J-  6)        sin  a  cos  b  -(-  cos  a  sin  b 
cos  (a  -f-  b)        cos  a  cos  b  —  sin  a  sin  b 

Dividing   both   terms   of  the   fraction  in  the   second 
member  by  cos  a  cos  6,  reducing,  and  recollecting  that 


CIRCULAR  FUNCTIONS.  87 

the  sine  of  an  arc  divided  by  its  co-sine   is  equal  to 
its  tangent,  we  have 

tan  a  -f  tan  b 


(e)     tan  (a  -f  6)  = 


1  —  tan  a  tan  b 


Hence,  The  tangent  of  the  sum  of  two  angles  is  equal  to 
the  sum  of  their  tangents,  divided  by  1  minus  the  product 
of  their  tangents. 

Dividing  (6)  by  (a),  and  reducing,  we  have 

cot  a  cot  b  —  1 


(/)     cot  (a  -f-  6)  = 


cot  a  4-  cot  b 


Hence,  The  co-tangent  of  the  sum  of  two  angles  is  equal 
to  the  product  of  their  co-tangents,  minus  1,  divided  by  the 
sum  of  their  co-tangents. 

Dividing  (c)  by  (d),  and  reducing,  we  have 

tan  a  —  tan  b 


(g)     tan  (a  —  b}  = 


1  -4-  tan  a  tan  b 


Hence,  The  tangent  of  the  difference  of  two  angles  is  equal 
to  the  tangent  of  the  first  minus  the  tangent  of  the  second, 
divided  by  1  pliis  the  product  of  their  tangents. 

Dividing  (c/)  by  (e),  and  reducing,  we  have 

cot  a  cot  b  -f  1 

(/O     cot  (a  —  6)  =  -    r-y—  -  • 

cot  b  —  cot  a 

Hence,  The  co-tangent  of  the  difference  of  two  angles  is  equal 
to  the  product  of  their  co-tangents,  plus  1,  divided  by  the  co- 
tangent of  the  second,  minus  the  co-tangent  of  the  first. 


94.    Problems. 

1.  Prove  that   (e),  (/),  (#),  (K)  become   homogeneous 
by  introducing  R. 


88  TRIGONOMETRY. 

2.  Deduce  (g)  from  (e)  by  substituting  —  6  for  b. 

3.  Deduce  (h)  from  (/)  by  substituting  —  6  for  6. 

4.  Deduce  (/)  from   (e)  by  taking   the   reciprocal  of 

each  member,  substituting  -- —  for  tan  o,      ,  t  for  tan  6, 

cot  a  cot  6 

and  reducing. 

5.  Deduce,  in  like  manner,  (A)  from  (g). 

6.  Find  the  value  of  sin  (a  -\-b  -{-  c)  by  substituting 
b  -{-  c  for  b  in  (a). 

7.  Find  the  value  of  cos  (a  -f-  b  -f  c)  by  substituting 
6  -f-  e  for  &  in  (6;. 

8.  Find  the  value  of  tan  (a  -\-  b  -\-  c)  by  substituting 
b  -j-  c  for  6  in  (e). 

9.  Find  the  value  of  cot  (a  -f  b  -f-  c)  by  substituting 
b  +  c  for  b  in  (/). 


95.   Functions  of  Double  and  Half  Angles. 

Making  b  =  a  in  (a),  (6),  (g),  and  (/),  we  have 

(1)  sin  2  a  =  2  sin  a  cos  a. 

(2)  cos  2  a  —  cos2  a  —  sin2  a. 

2  tan  a 

(3)  tan  2  a  =  r-       -^—  - 

1  —  tan2  a 

cot2  a  —  1 
(4      cot  2  a  =     0 

2  cot  a 

Substituting  $  a  for  a  in  (1),  (2),  (3),  (4),  we  have 

(5)  sin  a  =  2  sin  \  a  cos  J  a. 

(6)  cos  a  —  cos2  J  a  —  sin2  £  a. 


CIRCULAR   FUNCTIONS.  89 

2  tan  i  a 


2  cot  J  a 

Substituting  1  —  sin2  Ja  for  cos2£a,  then  1 — cos2-Ja 
for  sin2  Ja,  in  (6),  and  reducing,  we  have 

(9)     1— cos  a  =  2  sin2  %  a. 
(10)     1  +  cos  a  =  ^  cos2  ^  a. 


/UN     „:„  i /A — cos  a 

•   -     (11) 


*  2 

Dividing  (11)  by  (12),  then  (12)  by  (11),  we  have 

(13)  tan  ^  a  — -  \l  a . 

*  1  +  cos  a 

(14)  cot£a='J    +cos  a. 

*  1  —  cos  a 

Dividing   (5)   first  by  (10),  then   by  (9),  and   trans- 
posing, we  have 

(15)  tan  J  a  =  -    : 


8na 


1 

1  —  cos  a 

Taking  the  reciprocal  of  (16),  then  of  (15),  we  have 
1  —  cos  a 


(17)  tan  J  a  = 

(18)  cot  J  a  = 


sin  a 

1  -}-  cos  a 
sin  a 


Let   the   formulas   of   this    article    be   expressed    in 
words. 

S.  N.  8. 


90  TRIGONOMETRY. 

90.   Consequences  of  (a),  (b),  (c),  (d). 

Taking   the   sum   and   difference   of  (a)  and  (c),  (d) 
and  (6),  we  have 

(1)  sin  (a  -j-  6)  +  sin  (a  —  b)  =  2  sin  a  cos  6. 

(2)  sin  (a  -f  6)  —  sin  (a  —  b)  =  2  cos  a  sin  6. 

(3)  cos  (a  -f  6)  +  cos  (a  —  6)  —  2  cos  a  cos  6. 

(4)  cos  (a  —  b)  —  cos  (a  -)-  6)  —  2  sin  a  sin  6. 


Let    {a_b  =  dj       then       ft  =  ,  (g  __ 

Substituting  the  values  of  a  +  6,  «•  —  6,  a,  and  6,  in 
(1),  (2),  (3),  and  (4),  we  have 

(5)  sin  s  -\-  sin  d  =  2  sin  J  (s  +  d)  cos  J  (s  —  d). 

(6)  sin  .$•  —  sin  d  —  2  cos  ^  (s  -f-  d)  sin  %  (s  —  d). 

(7)  cos  s  -\-  cos  d  =?  2  cos  ^  (s  -f  d)  cos  J  (a  —  d). 

(8)  cos  d  —  cos  8  =  2  sin  -J  (s  -j-  d)  sin  J  (8  —  d). 

By  formula  (5)  of  the  preceding  article  we  have 

(9)  sin  (8  +  d)  =  2  sin  J  (s  -f  d)  cos  \  (s  -f-  d). 
(10)     sin  (s  —  d)  =  2  sin  i  («  —  d)  cos  J  (s  —  d). 

Dividing  each  of  these  formulas   by  each  of  the  fol- 
lowing, we  have 


sin^-j-sind     sin-|(s-t-d)  cosj(s — d)     tanj(8-|-d) 
(11) 


sins — sind  cosj(8-f-d)  sin£(8 — rf)      tan^(s— d) 

S_in^+sin_d  smi^+d)      tan 

cos  s  +  cos  a  cos  -J  (s  -f  a) 

_0,     sin. 5?  + sine?  cosj(s  —  d) 

(16)  — ^-=COl-?(S  — 

cos  a  —  cos  s  sin  J  (8  —  d) 

sin  s  -f  sin  d  cos  -J  (s  —  d) 

sin  (s  -f  d)  "     cos  £  (s  -f-  d) 


CIRCULAR  FUNCTIONS.  91 


(10) 

(16) 
(17) 
(18) 
(19) 
(20) 
(21) 

(23) 
(24) 
(25) 

sin  (a  —  d) 
sin  s  —  sin  d 

sin  J(a  —  c/) 
sin  -J  (a  —  d) 

tnr»I/e         /7^ 

cos  s  -f  cos  d 

cos  £  (a  —  rf) 

cos  d  —  cos  s 
sin  s  —  sin  d 

sin  3-  (s  -j-  d) 
sin£(a  —  d) 

cos  J  (a  -f-  d) 

sin  (a  -j-  d) 
sin  s  —  sirfrf 

sin  £(a  -|-  d) 

sin  (s  —  d) 
cos  a  -f-  cos  d! 

cos  J  (a  —  d) 
cot  J  (a  +  d) 

cos  d  —  cos  a 
cos  s  -f  cos  d 

tan-J(s  —  f?) 
cos  J  (s  —  rf) 

cos  s  -f-  cos  d 

sin  \  (a  -f  d) 
cos  J  (a  -f  d) 

sin  (a  —  f?) 
cos  d  —  cos  a 

sin  J(a  —  f?) 
sin  J  (a  —  d) 

sin  (a  -f-  d) 
cos  d  —  cos  s 

cos  J  (a  -f-  c?) 
sin  J  (a  -f  d) 

sin  (s  —  d) 

sin  (s  -f~  d) 

cos  -J  (a  —  d) 
sin  -|  (a  -I-  d) 

sin  (a^d) 

sin  J  (s  —  d) 

cos  J  (s  —  d) 

Formula  (11)  gives  the  proportion, 

sin  s  -f-  sin  d  :  sin  s  —  sin  d  :  :  tan  -J  (s  -f-  c?)  :  tan  £  (s  —  d). 

Hence,  The  sum  of  the  sines  of  two  angles  is  to  their 
difference  as  the  tangent  of  one-half  the  sum  of  the  angles 
is  to  the  tangent  of  one-half  their  difference. 

Let  us  apply  this  principle  in  solving  triangles 
when  two  sides  and  their  included  angle  are  given. 
Article  75. 


92  TRIGONOMETRY. 

a  :  b  ::  sin  A  :  sin  B. 

a-\-b  :  a  — 6::  sin  A  -{- 

sin  B  :  sin  A  —  sin  B. 


B 


sin  A+  sin  B  :  sin  A—  sin  B  : :  tau±(A+B)  :  tan±(A— B}. 
.'.     a  +  6  :  a  —  6  : :  tan  K^+£)  :  tan  %(A—B). 


97.   Theorem.. 


sum 


The  square  of  any  side  of  a  triangle  is  equal  to  the 
of  the  squares  of  the  other  sides,  minus  twice  their  product 
into  the  co-sine  of  their  included  angle. 

1st.  When  the  angle  is  acute. 
(1)     m  —  b  —  7i.  B 

(I)2  =(2)     m*  =  6*+n2— 26n. 

Zj» 

(3)    P2=p*.  * 


(2)+(3)=(4)     mz-f  p»  =  62  +  n2+  p2 ._  2  6w. 
But  m2  -f  p2  =  a2  and  n2-f  p2  —  c2,     .  • .  (4)  becomes 

(5)  a2  =  b2  -\-  c2  —  2  bn. 

But  n  —  c  cos  J,  which  substituted  in  (5)  gives 

(6)  a2  =  62  +  c2  —  2  6c  cos  A. 

2d.  When  the  angle  is  obtuse. 
(1)     m  =  b  +  n,  B 

(3)    p2=p2.  ^""^ — ^ — r 

(2) +  (3)  =  (4)     m2+P2- 


CIRCULAR  FUNCTIONS.  93 

But  m2-f  j32  —  a2  and  n2-\-p2  =  c2,    .*.  (4)  becomes 

(5)       ft2^2_!_c2_|_2  6tt. 

But  ?i  =  c  cos  iL4.D  —  —  c  cos  5^4  C  =  —c  cos  A 

.-.     (6)     a2  =  62+  c2—  2  6c  cos  A. 


98.   Problem. 

Tb  find  the  angles  of  a  triangle  when  the  sides  are  given. 
From  either  formula  (6)  of  the  last  article  we  have 

7)2     l     C2  _  a2 

/•«  \  A  I  ** 

(1)    cog^=      ___ 

Hence,  The  co-sine  of  any  angle  of  a  triangle  is  equal 
to  the  sum  of  the  squares  of  the  adjacent  sides,  minus  the 
square  of  the  opposite  side,  divided  by  twice  the  rectangle  of 
the  adjacent  sides. 

Formula  (1)  gives  the  natural  co-sine  of  A;  hence, 
A  can  be  found.  But  it  is  best  to  place  the  formula 
under  such  a  form  as  to  adapt  it  to  logarithmic  com- 
putation. 

Adding  1  to  both  members  of  (1)  we  have 

(I  +  g)2_a2  __  (a  +  6-f-C)(6  +  g  —  a) 

Tbc~  2  be 

But  1  +  cos  A  =  2  cos2  £  A.     Article  95,  (10). 

(a  +  b  +  c}(b  +  c—  a) 
Let  «+&+*=;>,  then  -          -- 


Substituting  these  values  in  (2),  and  dividing  by  2, 
we  have 


94  TRIO  ONOMETE  Y. 

In  like  manner,  (5)     cos  £  B  = 
Also,  (6)    cos  $  C  = 


-— 
06 

Introducing   R,   applying   logarithms,  and   reducing, 
(4)  becomes 

log  cos  \A  —  \  [log  Jp+log  typ  —  a)  -\-a.c.  log  b-\-a.c.  log  c]. 

In   like   manner   introduce   R  and  apply  logarithms 
to  (5)  and  (6). 

By  subtracting  ..both  members  of  (1)  from  1  and  re- 
ducing we  find 


be 
(8)     sin  J  B  = 


(9)     sin^C  = 

* 


(7)  .,_  (4)  =  (10)    tan  \  A  = 


(9)  ^_  (6)  =  (12)     tan  i  C  = 


99.   Examples. 

r  a  =125.  ^|  r^  =  55°46'  18". 

1.  Given  1  b  =  150.    >  Required  <  5=  82°  49'  08". 

I  c  =  100.  J  I C  =  41°  24'  34". 

ra=864.   -|  ryl^41°  00' 38". 

2.  Given  1  b'^  1308.  V  Required  <  B  ^'83°  25'  14". 

U  =  1086.J  (c— 55°  34' 08". 


CIRCULAR  FUNCTIONS.  95 

100.  Problem. 

To  find  the  area  of  a  triangle  when  two  sides  and  their 
included  angle  are  given. 

Let  k  denote  the  area  of  the  tri- 
angle  ABC,  of  which  the  two  sides 
6  and  c  and  their  included  angle  A 

are  given.  ° 

(1)  2  k  =  bp. 

(2)  p  —  c  sin  A. 
.'.     (3)     2  k  =  be  sin  A. 

Introducing  R,  and  applying  logarithms,  we  have 
log  (2  k)  =  log  b  -f  log  c  -f  log  sin  4  —  10. 

101.  Examples. 

1.  Two  sides  of  a  triangle  are  345.6  and  485,  respect- 
ively, and   their   included   angle  is  38°  45'  40";    what 
is  the  area?  Am.  52468. 

2.  Two  sides  of  a  triangle  are  784.25  and  1095.8,  re- 
spectively,   and    their   included   angle    is    85°  40'  20"; 
what  is  the  area.  Ans.  428470. 

102.   Problem. 

To  find  the  area  of  a  triangle  when  the  three  sides  are  given. 
By  the  last  problem  we  find 

(1)  k  =  \  be  sin  A, 

(2)  sin  A  =  2  sin  J  A  cos  J  A.     Article  95,  (5). 


(3)     sin  J  A  =  —         P  —  c  .  Article  98,  (7). 

be 


96  '  TRIGONOMETRY. 


(4)  cos  \A  =  ^/my-  Ify .  Article  98,  (4). 

(5)  sin     A  = 


be 


.  •  .      (6)  k  =  V  \p(\p  —  a)  (ip  —  6)  (ip  —  r  ). 

103.    Examples. 

1.  The  sides  of  a   triangle   are   40,  45,  55,  required 
the  area.  An*.  887.412. 

2.  The  sides  of  a  triangle  are  467,  845,  756,  required 
the  area.  Ans.  175508. 

104.    Problem. 

Given   the  perimeter  and   angles  of  a  triangle,   required 
the  sides. 


. 
a        sin  A 

Adding  and  reducing  by  Articles  96,  (5)  and  95,  (5), 
we  have 

b      c      sin 


— 
a  sin  J  A  cos  J  ^4 

sin  %(B  -h  C)  —  cos  £  4,    and  sin  J  ^4  —  cos  |  (5  +  C). 


a          cc 
Adding  1  to  both  members,  we  have 

cos  4  (B—  C) 


a  cos  £  (J5  -f  C) 

Let  _p  =  a  -j-  6  -f  c,  and  reduce  by  96,  (7),  we  have 
2  cos  i  5  cos 


C5)    -.  — 


sin  \A 


.  • .     (6)      a  =     ^P  sin  -M 

cos  ^5  cos  &  C  ' 


CIRCULAR  FUNCTIONS.  97 

Introducing  R  and  applying  logarithms,  we  have 

log  a  =  log  \p  +  log  sin  J  A  -f 

a.  c.  log  cos  J-  5  -f-  «••  c.  log  cos  J  C  —  10. 

Similar  formulas  can  be  found  for  6  and  c.  But, 
after  a  is  found,  6  and  c  can  be  more  readily  found 
by  article  69. 

105.    Examples. 

1.  Given  p  =  150,    ,4  =  70°,    72^60°,    (7=50°,    re- 
quired «,  6,  c. 

Ans.  a  =  54.81,.  6  =  50.51,  <?  ±=  44.68. 

2.  Given   ;>  ==  31234.36,   A  ==  35°    45',    5  =  45°    28', 
(7=98°  47',  required  a,  ,6,  c. 

.  a  =  7985,  6  =  9742.5,  c  ^  13506.86. 


3.  Given  p  =  375,  A  &  55°  46'  18",  B  £=  82°  49'  08", 
7—41°  24'  34",  required  a,  6,  c. 

Ans.  a  =  125,  6  =  150,  c  =  100. 


106.    Problem. 

Given  tfie  three  sides  of  a  triangle,  to  find  the  radius  of 
the  inscribed  circle. 

(1)     BOC+AOC+AOB  = 

(2) 
(3) 

(4)     AOE—\cr. 
.'.     (5) 


But  (6) 

S.  N.  9. 


98  TRIG  OS 0 METE  Y. 

.'.     (7)     \pr  —  l/~ 


r  -_ 


\ 


107.  Examples. 

1.  The  three   sides   of   a   triangle    are   20,  30,  40,  re- 
spectively, required  the  radius  of  the  inscribed  circle. 

Am.  6.455. 

2.  The  three  sides  of  a  triangle  are  100,  150,  200,  re- 
spectively, required  the  radius  of   the  inscribed  circle. 

Am.  32.275. 

108.  Problem. 

Given  the  three  sides  of  a  triangle  to  find  the  radius  of 
the  circumscribed  circle. 

Let  0  be   the   center  of  the   circle, 
and  R  the  radius. 

Let  OD  be  perpendicular  to  6,  then      , 

A  A 


.     , 

The  angle  0  =  the  angle   B,  since  each  is  measured 
by  one-half  the   arc    AC. 

(1)     AD  =  4-  =  AO  sin  0  =  R  sin  B. 

2i 

.'.     (2)     R  = 


2  sin  B 
2 


sin  B  =  2  sin  ^B  cos  ^B  = 

ac 

,£.     R=   abc =  abr 

4  V\p  (4p  —  a)  (\p  —  b}  ( \ p  —  c)        4  k 

Prove  that  the  formula  will  be  the  same  if  the  cen- 
ter is  without  the  triangle. 


CIRCULAR  FUXCTIOXS.  '  99 

109.  Examples. 

1.  The  sides  of  a  triangle  are   7,  9,  10,  respectively, 
required  the  radius   of  the   circumscribed  circle. 

Ans.  5.148. 

2.  The  sides  of  a  triangle  are  50,  60,  70,  respectively, 
required  the  radius  of  the  circumscribed  circle. 

Ans.  35.72. 

110.  Theorem. 

The  perpendicular  let  fall  on  either  side  of  a  triangle  from 
the  vertex  of  the  opposite  angle  is  equal  to  that  side  into  the 
product  of  the  sines  of  the  adjacent  angles  divided  by  the 
sine  of  the  sum  of  those  angles. 

(1)    p  =  c  sin  A. 

(2)     sin  B  :  sin  C  :  :  b  :  c,     .'.     c=-    — =- • 

sin  jj 

b  sin  A  sin  C 


(4)     sin  B  =  sin  [180°  —  (A+  C)]  == 
sin  04  +  C). 


_ 
p' 


111.    Problem. 

Given  ^0  ^ree  sid<?s  o/  a  triangle  to  find  the  radii  of 
the  escribed  circles. 

The  escribed  circles  are  the  three  circles  external  to 
the  triangle,  each  tangent  to  one  side  and  to  the  pro- 
longation of  the  other  sides. 


100 


TRIGONOMETRY.- 


The  centers  .of  the  escribed  circles  are  the  points  of 
intersection   of   the   lines 
bisecting      the      external 
angles. 

The  radii  r,  r",  r'",  of 
the  escribed  circles,  will 
be  the  perpendiculars  let 
fall  from  their  centers  0', 
0",  0'",  respectively,  on 
the  three  sides  a,  6,  c. 

Hence,  by  the  last  ar- 
ticle, 

,_  a  sin  (90° 
'~~ 


/ox          > 

.  •  .    (2)    r  '  — 


a   COS 


COS 


Substituting  the  value  of  tan  4^4>  article  98,  we.  have 


1 1 '2.    Examples. 

1.  Given  the  sides  of  a  triangle,  6,  9,  11,  required  the 
radii  of  the  three  escribed  circles. 

Am.  3.854,  6.745,  13.49. 

2.  Given  p  =  100,  .4^55°,  £  =  60°,  0=65°,  required 
the  radii  of  the  three  escribed  circles. 

[See  (2),  Art.  111.]  Ans.  26.028.  28.867:  31.854. 


CIRCULAR  FUNCTIONS.  101 

113.   Theorem. 

The  product  of  the  radius  of  the  inscribed  circle  and  the 
radii  of  the  three  escribed  circles  is  equal  to  the  square  of 
the  area  of  the  triangle. 

The  product  of  (8),  article  106,  and  (3),  (4),  (5), 
article  111,  gives 

H  1.4 

r  rVV" _  __ __ ,          _   __  1U2 

~ 


114.   Theorem. 

The  reciprocal  of  the  radius  of  the  inscribed  circle,  the 
sum  of  the  reciprocals  of  the  radii  of  the  escribed  circles, 
and  the  sum  of  the  reciprocals  of  the  perpendiculars  let  fall 
from  the  vertices  of  the  three  angles  on  the  opposite  sides  of 
a  triangle  are  equal  to  each  other. 

Taking  the  reciprocal  of  (8),  article  106,  we  have 

m     !  -    p 
~~  2k' 

Taking  the  sum  of  the  reciprocals  of  (3),  (4),  (5), 
article  111, 

111     _p-2a      p-2b      p-2c       p 

W   -7-  --  r»  -  rn,  -  -gy-     -gj-       2k     ~2k- 

Let  p',  p",  p'",  respectively,  be  the  perpendiculars  let 
fall  from  the  vertices  of  the  three  angles  on  the  sides 
a,  b,  and  c.  Then  we  have 


In  like  manner,  -^  =  —  •     Also,  -777-  ==  - 


102  TRIGONOMETRY. 

m     —         —         —         a+b  +  c  _     _P__ 
p'          p"  "  p'"  ~~          2k          ~~  2  k  ' 

...  (4)  J-^J^  _i_  + 4.=^  + j_  +  4 


115.   Problem. 

To  find   the  disfyrtce  between  the  centers  of  the  circum- 
and  "instnfad  circles  of  a  triangle. 


?  'iiid'r  b<?  thj&  radii,  and  P 
and  0  the  centers  of  the  circles,  and 
let  D  =  OP. 

Draw  PE  perpendicular  to  AC.    The 
angle  APE  =  B,  since  each  is  meas- 
ured by  one-half  the  arc  AC;   but  PAE  =  90°  —  APE, 
.-.  PAE=90°—B.    OAC=1>A.    PAO=PAE—OAC. 


PAO  =  90°  —  B  —  \A  =  4(C- 


(1)     OP2=^4P2-M#2—  2  APxAO  cos  PAO.     Art.  97. 

Substituting    the  values  of  OP,    AP,    AO,  and  PAO, 
we  have 


6          _  6  _  (  108,  (2). 

~  2  sin  B  "  4  sin  J£  cos  JB  '          3'  1    95,  (5). 

6  sin  \A  sin  \C      b  sin  J^4  sin  JC      A 
-  ~ 


r2  4  J?r  sin  |P  sin 

sin2  W  ~"  sin  I  A 


CIRCULAR  FUNCTIONS. 


103 


Substituting  in  (2),  and  reducing  by  article  91,  (d), 
and  89,  (6),  we  have 


Dt=Bf_ 


sm 


= 


,-.     (7)     D  =  VRt  —  2  Rr. 


116.    Examples. 

1.  The   sides   of  a   triangle  are   12,  13,  15;    required 
the  distance  between  the  centers  of  the  circumscribed 
and  inscribed  circles.  Ans.  1.616. 

2.  Two  sides   of  a  triangle  are  35  and  37,  and  their 
included   angle   is  50°;    required  the  distance  between 
the  centers  of   the  circumscribed  and  inscribed  circles. 

Am.  3.266. 

3.  The  perimeter  of  a  triangle  is  120,  the  angles  are 
40°,  60°,  and   80°,  respectively;    required   the   distance 
between  the  centers  of  the  circumscribed  and  inscribed 
circles.  Ans.  8.353. 


117.    Problem. 

To  find  the  distance  between  the  centers  of  the  circumscribed 
and  escribed  circles. 


Let  /,  r",  r"  be  the 
radii  of  the  escribed 
circles,  and  £>',  D",  D'", 
be  the  distances  of 
their  centers,  0',  0", 
0'",  respectively,  from 
P,  the  center  of  the 
circumscribed  circle, 
whose  radius  is  R. 


104  TRIG  ONOMETR  Y. 

As  in  the  last  Problem,  we  find 


sin2  \A  sin  \A 

(*)     R=        a       -  "  Arts  I108'®' 

2  sin  ,4     4  sin  \A  cos  A<4  ''I   95,  (5). 


(3)    ^co 


Substituting  (4)  in  (1),  and  reducing  by  (d)  and  (6), 
we  have 


(5)    iy.= 


.-.     (6)     #  = 

.-.     (7)     D"  = 
.-.     (8)     D"'= 


118.    Examples. 

1.  The  three,  sides  of  a  triangle  are  21,  23,  26;    re- 
quired  the   distances   from   the   center  of  the   circum- 
scribed   circle    to    the    centers    of    the    three    escribed 
circles.  Ans.  25.19,  26.64,  29.73. 

2.  The   angles   of  a   triangle   are   56°,    60\    64°,   the 
greatest  side   is   25 ;    required   the    distances    from   the 
center  of  the  circumscribed  circle  to  the  centers  of  the 
three  escribed  circles.  Am.  26.96,  27.80,  28.65. 

3.  Given   p  ==  100,    A  =  55°,    B  =--  60°,     C  =£  65°, 
required  U,  D".  D"'.  Ans.  37.10,-  38.55,  40.01. 


CIRCULAR  FUNCTIONS. 


105 


119.   Problem. 

To  find  the  distance  between  the  centers  of  the  inscribed, 
and  escribed  circles. 


Let  Dj,  D2,  £>3,  be  the 

distances. 

In  the  triangle  OO'E, 
we  have 

r'  —  r 
(1)     *>!=;- 


sin  \A 


Substituting  the  values  of  r',  r,  and  sin  \A,  we  have 


120.   Examples. 

1.  The   three   sides  of  a  triangle  are   30,  50,  60;    re- 
quired  the   distances   between    the   centers   of   the    in- 
scribed and  escribed  circles.       Ans.  31.05,  56.69,  87.83. 

2.  The  sides  of  a  triangle  are  500,  600,  700;    required 
the  sides  of  the  triangle  formed  by  jpining  the  centers 
of    the    inscribed    and    circumscribed    circles    and   the 
center  of   the  escribed  circle,  tangent   to   the  sides  600 
and  700  produced.  Ans.-  540.06,- 104.58,  624,58, 


10(5  r/,'  /f.-n.YM  v  rrnv. 

is  Kxercises. 


i.  Prove  bhfctsin  ir)°  =       ~       cos  I5°  = 


21/2'  2  V  2 

tan  16°  =  2  —  V  li^     cot  15°  =  2  -f  V  ~Z,     sec  r> 
f} 
I 
2.   Find   the  sine  and  00  line  of  75°. 

J,,,  sin75°=i^l±i,    cos  75-=  *I      ' 
21    2  LM    2 

8.  Why  is  sin  76°=  cos  15°,  and  cos  75°=-  sin  15°  ? 

I     How    may    the    value*    of    tangent,    co-tangent,    se- 


t,  and    eo-seeant    of    7-~> '    he    found    from    the   values 
of  the  sine  and  co-sine? 

•r>.  Find  the  functions  of  150°. 

Am.  sin  I/XT  =  4,  cos  150°  = 
0.  (Jiven  sin  n  +  cos  n  —  172,  to  find  a. 

.l*w.  45°,  or  45° 4-  300°;  or,  in  general,  ''    \   L>  TH. 
7.  (Jiv<«n  sin  2  <f        eos  </,  to  find  a. 

7T 

•  l"-s'.        i   2  ?m,  or  f  tr   j  2  TT?L 

S.    Prove    that    the  sum  of    the   tangents  of   the    thn-e 
anglea  of  a  plane  triangle  is  i><|ii:il   to  their  produet. 

1).    Pr»»ve    tliat    the   sum   of  the  cotangents  of  one-half 
the  angles  of  a  plane  triangle  is  e^ual  to  llu»ir   product. 

10.  Prove    that    Mir   is    isoseeles    if    COS      I 

2  sin  r 

11.  I'rove   that    the  sum  of    the  diameters    of   the    in- 
serihed    and    eirenm-vnhed    eireles    of    any    plane    tri- 
angle ABC  is 

a  cot  A  4  l>  cot  B  4  c  cot  C. 


107 

12.  If  ft  is  ihe  hase  of  the  triangle  Al\<\  '/>,  the  per- 
pendiCUl&T  ti»  the  h;ise  IVom  the  vertex  of  the  opposite 
anj.de,  and  *,  the  sum  of  the  sides  H.  and  r,  prove  that 

tan  i*  -  2  '"' 


115.    If  />   is  tli<>   f»asc  of  Uic   triangle    AIM',  />,   UK-    )MT 
|.cndicidar    to    tin:    hasc    from    (lie    vertex    of    the  oppo- 
site;   an^lc,  ;md    J,   the   dillerenee  of    tin-  sides  a  and  r, 
{•rove  that 

.B_(6  +  d)(ft-«l) 

2  /,,, 

14.  If  a,  />,  and  r  l»e  the  sides  <•('  the  triangle  ABC, 
x,  the  sum  of  the  flidew  a  and  r,  and  r,  the  radius  of 
the  inscribed  eirde,  prove  tliat 

o  r 


122.   Computation  of  Natural  Fiinctioiw. 


the    length  of  the  senii-eireiinifei-enee  to  the 

radius  1,  which  JH  n  =  3.14  \KWWM\WK  .  .  .  hy  1080, 
the  niiiiiher  of  ininuteH  in  ISO1,  the  <jiiof,ienf,  which  is 
.000'2(.K)SSS-2  .  .  .  ,  will  he  the  length  of  the  are  T,  and 
will  differ  insensihly  from  its  sine. 

.',     (1)     sin   1'       .(X0290SSS2. 

.-.     (2)     COM  1'       ll        Hin54  1'        WH9999577. 


Adding   (a)  and    ((?),  then    <  I,  )   and   (<l  >,  articles  89,  91, 

'  and  transposing, 


(3)  sin  (a  +  b)  --  2  sin  <>  col  //  —  sin  (a  —  It). 

(4)  cos  (a  +  6)  ~  2  cos  a  cos  6  —  cos  (a  —  &;. 


108  TRIGONOMETRY. 

If  in  (3)  and  (4)  6  =  1,  a  =  1,  2,  3  . . . ,  in  succession, 
we  have 

sin  2'  =  2  cos  1'  sin  1'—  sin  0'=  .0005817764. 
sin  3'  =  2  cos  1'  sin  2'  —  sin  1'  ==  .0008726646. 
sin  4'=  2  cos  1'  sin  3'  —  sin  2'  =  .0011635526. 

cos  2'  =  2  cos  1'  cos  1'  —  cos  0'  ±±=  .9999998308. 
cos  3'  =  2  cos  1'  cos  2'  —  cos  1'  ±=  .9999996193. 

To  facilitate  computation,  for  2  cos  1'  =  1.9999999154, 
use  its  equal,  2  —  .0000000846.  Then  we  have 

sin  2'  =  2  sin  V  —  .0000000846  sin  1'—  sin  0'. 
sin  3'  =  2  sin  2'  -  .0000000846  sin  2'—  sin  1'. 

After  finding  the  sines  and  co-sines,  the  tangents  and 
co-tangents  can  be  calculated  from  the  formulas: 

/K,  sin  a  cos  a 

(5)     tan  a  =  -     —  •       (6)     cot  a  =  -      —  • 
cos  a  sm  a 

It  is  not  necessary  to  carry  the  computation  beyond 
45°,  since  sin  «— cos  (90°  —  a),  etc. 

The  logarithmic  functions  can  be  found  from  the 
corresponding  natural  functions  by  the  method  of 
article  60. 


SPHERICAL   TRIGONOMETRY. 

123.    Definition  and  Remarks. 

Spherical  Trigonometry  is  that  branch  of  Trigonome- 
try which  treats  of  the  solution  of  spherical  triangles. 

If  any  three  of  the  six  parts  of  a  spherical  triangle 
are  given,  the  remaining  parts  can  be  computed. 

The  radius  of  the   sphere   is  taken   equal  to  1,  and 


RIGHT  TRIAKULES.  109 

each    side    has    the    same    numerical    measure    as   the 
subtended  angle  whose  vertex  is  at 
the  center  of  the  sphere.     Thus, 

a  L  BOO,  b  =  AOC,  c  =  AOB. 

An  angle  of  a  spherical  triangle 
is  the  angle  included  by  the  planes  of  its  sides  which 
is  measured  by  the  angle  included  by  two  lines,  one 
line  in  one  plane,  the  other  in  the  other,  both  per- 
pendicular to  the  common  intersection  of  the  planes 
at  the  same  point. 

Thus,  if  BE,  in  the  plane  AOB,  is  perpendicular  to 
OA,  and  if  ED,  in  the  plane  AOC,  is  perpendicular  to 
OA,  then  the  angle  BED  will  measure  the  inclination 
of  the  planes  AOB  and  AOC,  and  will  be  equal  to  the 
angle  A  of  the  spherical  triangle. 


RIGHT   TRIANGLES. 
124.   Napier's  Circular  Parts. 

Napier's  circular  parts  are  the  two  sides  adjacent  to 
the  right  angle,  the  complements  of  their  opposite 
angles,  and  the  complement  of  the  hypotenuse. 

Thus,  if  HBP  is  a  spherical 
triangle,  right-angled  at  H,  the 
circular  parts  are  b,  p,  90°  —  B, 
90°  — P,  and  90°—  h.  9()0_p. 

Adjacent  parts  are  those  which 
are  not  separated  by  an  intervening  circular  part. 

Thus,  b  and  90°  — P,  90°  — P  and  90°  — /*,  9<P—  h 
and  90°  —  B,  90°—  B  and  p,  p  and  6  are  adjacent 
parts. 

The  right  angle  H  is  not  regarded  as  a  circular 
part,  nor  as  separating  the  parts  6  and  p. 


110  TRIG  ONOMETR  Y. 

Opposite  parts  are  those  which  are  separated  by  an 
intervening  circular  part. 

Thus,  b  and  90°—  A,  90°  — P  and  90°  —  P,  90°  —  h 
and  p,  90°— #  and  6,  p  and  90° — P  are  opposite  parts. 

Any  one  of  these  five  circular  parts  is  adjacent  to 
two  of  the  remaining  parts,  and  opposite  the  other 
two  parts. 

Of  any  three  circular  parts,  one  part  is  either  adja- 
cent to  both  the  others  or  opposite  both. 

A  middle  part  is  that  which  is  adjacent  to  two  other 
parts,  or  opposite  two  other  parts. 

125.   Exercises. 

Tell  which  is  the  middle  part,  and  whether  the  other 
parts  are  adjacent  to,  or  opposite,  the  middle  in  the 
following  : 


1.  90°— B,  90°— P,  9(T— 

2.  6,  90°—  h,  p. 

3.  90°—  h,  90°-£,  p. 

4.  90°— P,  9(T— 5,  6. 

5.  6,  90°— 5,  p. 


6.  90°— P,  90° 


7.  b,  90°— P,  p. 

9.  90°—  h,  90°— P,  b. 
10.  90°— P,  90°— £,  p. 


126.    Napier's  Principles. 

1.   The  sine  of  the  middle  part  ix  equal  to  the  product 
of  the  tangents  of  the  adjacent  parts. 

Draw  BD  and  DE,  respectively 
perpendicular  to  OH  and  OP,  and 
draw  BE.  BDE  is  a  right. angle, 
since  the  plane  BOH  is  perpendicu- 
lar to  the  plane  POH,  and  BD  is 
perpendicular  to  OH.  The  angle  BED  is  equal  to  P. 


EIGHT  TRIANGLES.  Ill 

EB  =  sin  h,  OE  =  cos  7i,  DB  =  sin  p,  and  OD  =  cos  jy. 

ED        OE        ED 

"Fir  X  -7TTT'    or  cos  P=  cot  /i  tan  6. 


.-.     (1)     sin  (90°—  P)  ==  tan  (90°—  h~)  tan  6. 

ED        DB        ED 

or  Bin  6  =  tan   >  cot  P. 


.  •  .     (2)     sin  b  =  tan  p  tan  (90°  —  P). 

By  changing  P,  6,  p  into  B,  p,  b,  (1)  and  (2)  become 

(3)  sin  (90°—  5)  =  tan  (90°—  A)  tan  p. 

(4)  sin  _p  =  tan  b  tan  (90°—  5). 

Multiplying  (2)  by  (4),  member  by  member,  we  have 
sin  b  sin  p  =  tan  6  tan  p  tan  (90°—  B)  tan  (90°—  P). 

Dividing  by  tan  b  tan  p,  and  reducing,  we  have 

cos  b  cos  p  =  tan  (90°—  5)  tan  (90°—  P). 
cos  6  cos  p  =  cos  EOD  xOD=OE=  cos  h  =  sin  (90°—  A). 
.  •  .     (5)    sin  (90°—  h)  i  tan  (90°—  P)  tan  (90°—  P). 

2.    JTie  siW  of  ffo  middle  part  is  equal  to  the  product  of 
the  co-sines  of  the  opposite  parts. 

OE  =  cos  EOD  X  -OD,  or  cos  h  =  cos  6  cos  p. 
.  •  .     (6)     sin  (90°—  h)  =  cos  6  cos  p. 

DB=EB  sin  DEB,  or  sin  p  =  sin  h  sin  P. 

.  •  .     (7)     sin  p  =  cos  (90°—  h)  cos  (90°—  P).  ' 

sin  (90'—  A)  sin  p 
(3)     gives  sin  (90°-*)  =  _—  -^^-^  - 


112  I  RIG  ONOMETR  Y. 

This,  by  substituting  cos  6  cos  p  for  sin  (90° — A), 
cos  (90° — h)  cos  (90° — P)  for  sin  p,  and  reducing,  gives 

(8)  sin  (90°— R)  ~  cos  b  cos  (90°— P). 

By  changing  p,  P,  B,  b  into  6,  5,  P,  p,  (7)  and  (8) 
become 

(9)  sin  b  =  cos  (90°—  h)  cos  (90°— P). 
(10)     sin  (90°— P)  =cos  p  cos  (90 >— 5). 

These  ten  formulas  are  thus  reduced  to  two  princi- 
ples, from  which  the  formulas  can  be  written. 

The  memory  will  be  further  aided  by  observing  the 
common  vowel  a  in  the  first  syllables  of  the  words 
tangent  and  adjacent  of  the  first  principle,  and  the 
common  vowel  o  in  the  first  syllables  of  the  words 
co-sine  and  opposite  of  the  second  principle;  that  is, 
we  take  the  product  of  the  tangents  of  the  parts 
adjacent  to  the  middle,  and  the  product  of  the  co-sines 
of  the  parts  opposite  the  middle. 


127.  Mauduit's  Principles. 

If  we  take,  as  circular  parts,  the  complements  of 
the  two  sides  adjacent  to  the  right  angles,  their  oppo- 
site angles,  and  the  hypotenuse,  we  can  readily  deduce 
from  the  diagram,  or  from  Napier's  principles,  the 
following  principles: 

1.  The  co-sine  of  the  middle  part  is  equal  to  the  product 
of  the  co-tangents  of  the  adjacent  parts, 

2.  The  co-sine  of  the  middle  part" is  equal  to  the  product 
of  the  sines  of  the  opposite  parts. 

Let  the  ten  formulas  be  written  and  compared  with 
those  of  the  last  article. 


RIGHT  TRIANGLES. 


113 


128.  Analogies  of  Plane  and  Spherical  Triangles. 

The  formulas  which  demonstrate  Napier's  principles 
may  be  placed  under  forms  which  will  exhibit  the 
analogies  existing  between  Plane  and  Spherical  Tri- 
angles, as  in  the  subjoined  table. 


Plane  Right  Triangles. 


Spherical  Right  Triangles. 


1.  sin  P=-f-- 

ri 

1. 

sin 

p  =  ^^P  . 
sin  h 

2.  sin  B=*HJL 

2. 

sin 

B==  sin  ^ 

sin  h 

3.  cos  P  =  4-' 

A 

3. 

cos 

p       tan  b 
tan  h 

4^^*       D    -» 

4. 

cos 

P       tan  p' 

.    COo    Jj  j  "    • 

n 

tan  ^ 

5.  tan  P=-£-- 

0 

5. 

tan 

p       tanj) 
~  sin  b 

6.  tan  £=-—.. 
£> 

7.  sin  P  —  cos  5. 

6. 

7. 

tan 
sin 

„        tan  b 
sin  jo 

p  _  °Plj?  j 
cos  6 

8_f  »-.       T">                               7~> 

8. 

sin 

£       cos  P 

.  sin  B  —  cos  r. 

COS  JO 

9.         A.  =  6^  +  p«. 

9. 

cos 

^  =  cos  b  cos  p. 

10.           1  =  cot  5  cot  P. 

10. 

cos 

A  —  cot  B  cot  P 

These   formulas   can   be  committed   and   applied   in- 
stead of   Napier's    principles    by   those    who    prefer    to 
do  so.     The  analogies  will   assist  the  memory. 
S.  N.  10. 


1 14  TRIG  ONOMETR  Y. 

129.   Species  of  the  Parts. 

Two  parts  of  a  spherical  triangle  are  of  the  same 
species  when  both  are  less  than  90°  or  both  greater 
than  90°. 

Two  parts  of  a  spherical  triangle  are  of  different 
species  when  one  part  is  less  than  90°  and  the  other 
part  greater  than  90°. 

We  shall,  at  present,  consider  those  triangles  only 
whose  parts  do  not  exceed  180°. 

Let  it  be  remembered  that  the  sine  is  positive  from 
0°  to  180°,  and  that  the  co-sine,  the  tangent,  and  the 
co-tangent  are  positive  from  0°  to  90°,  and  negative 
from  90°  to  180°.  Hence,  if  the  co-sines,  tangents,  or 
co-tangents  of  two  parts  have  like  signs,  these  parts 
will  be  of  the  same  species;  if  they  have  unlike  signs, 
these  parts  will  be  of  different  species. 

„      cos  B        ,     .      „       cos  P 

sin  P  = r  and  sin  B  —  ~     -•     Art.  128,  7,  8. 

cos  o  cos  p 

Since  neither  P  nor  B  exceeds  180°,  sin  P  and  sin  B 
are  both  positive;  hence,  cos  B  and  cos  b  have  like 
signs,  so  also  have  cos  P  and  cos  p.  Therefore,  B  and 
b  are  of  the  same  species;  so  also  are  P  and  p. 

Hence,  The  sides  adjacent  to  the  right  angle  are  of  the 
same  species  as  their  opposite  angles. 

cos  h  =  cos  b  cos  p.     Art.  128,  9. 

If  h  <  90°,  cos  h  is  positive;  hence,  cos  b  cos  p  is 
positive ;  .-  • .  cos  6  and  cos  p  have  like '  signs ;  .  • .  b 
and  p  are  of  the  same  species;  .  •.  B  and  Pare  of  the 
same  species. 

Hence,  If  the  hypotenuse  is  less  than  90°,  the  two  sides 
adjacent  to  the  right  angle  are  of  the  same  species;  so  also 
are  their  opposite  angles. 


RIGHT  TRIANGLES.  115 

If  h  >  90°,  cos  h  is  negative;  hence,  cos  b  cos  p  is 
negative ;  . ' .  cos  6  and  cos  p  have  unlike  signs ;  .  • .  b 
and  p  are  of  different  species ;  .  • .  B  and  P  are  of 
different  species. 

Hence,  If  the  hypotenuse  is  greater  than  90°,  the  two 
sides  adjacent  to  the  right  angle  are  of  different  species;  so 
also  are  their  opposite  angles. 

Let  us  now  investigate  the  case  lt^JZ — ^ 

in  which   a  side   adjacent    to   the      p< 
right  angle  and  its  opposite  angle 
are  given. 

Let  p  and  P  be  given.  Produce  the  sides  PH  and 
PB  till  they  meet  in  P'.  The  angles  P  and  P'  are 
equal,  since  each  is  the  angle  included  by  the  plane 
of  the  arcs  PHP'  and  PBP'.  Take  P'H'  --=PH=b  and 
P'B'  =  PB=h.  The  two  triangles,  PHB  and  P'H'B', 
have  the  two  sides  PH  and  PB  and  the  included  angle 
P  of  the  one,  equal  to  P'H'  and  P'B'  and  the  included 
angle  P'  of  the  other;  hence,  they  are  equal  in  all 
their  corresponding  parts;  .*.  H'  =  H,  B' =  B,  and 
H'B'-=HB.  But  H  is  a  right  angle;  .  • .  H'  is  a 
right  angle.  Hence,  either  triangle,  PHB  or  PH'B', 
will  answer  to  the  given  conditions. 

Since  P'H'  and  PH  are  equal,  and  P'H'  and  PH' 
are  supplements  of  each  other,  PH  and  PH'  are 
supplements  of  each  other.  In  like  manner  it  can 
be  shown  that  PB  and  PB'  are  supplements  of  each 
other. 

When,  therefore,  a  side  adjacent  to  the  right  angle 
and  an  opposite  angle  are  given,  there  are  apparently 
two  solutions.  The  conditions  of  the  problem,  how- 
ever, may  be  such  as  to  render  the  two  solutions 
possible,  reduce  them  to  one,  or  render  any  solution 
•impossible. 


116  TRIG  ONOMETRY. 

Let  us   now  proceed  to  investigate  these  conditions. 

1.  When  P  <  90'  and  p  <  P.  ~T\ 

T<\       r          /p 
We   have   from  Napier's   princi-        ^x/     vjx^ 

1  TTT TLJ^ 

pies, 

sin  b  =  tan  p  tan  (90°  —  P),  or  sin  b  =  tan  p  cot  P 

Since  P  <  90°  and  p  <  P,  tan  _p  <  tan  P;  but  we 
have  tan  P  cot  P  — -  1 ;  .  • .  tan  p  cot  P  <  1 ;  hence, 
sin  b  <  1;  then  b  <  90°  or  6  >  90°;  hence,  b  may 
be  either  of  the  supplementary  arcs  PH  or  PH'  which 
have  the  same  sine  equal  to  tan  p  cot  P 

If  b  <  90°,  since  p  <  90°,  h  <  90° ;  if  6  >  90°, 
since  p  <  90°,  A  >  90°.  Hence,  if  P  <  90°  and 
p  <  P,  either  triangle,  PHB  or  P#'£',  will  satisfy  the 
conditions,  and  there  will  be  two  solutions. 

2.  When  P  <  90°  and  p  =  P. 

P 
We  have  sin  b  —  tan  p  cot   P, 

as  before. 

Since  p  =  P,  tan  jo  cot  P=  tan  P  cot  P  —  1 ;  there- 
fore, sin  6  =  1 ;  .  • .  6  =  90°,  or  PH  =  90°. 

From  Napier's  principles,  we  have 

sin  (90° —  h)  —  cos  6  cos  jo,  or  cos  h  =  cos  b  cos  £>. 

Since  b  —  90°,  cos  6  —  0 ;  .  • .  cos  b  cos  p  =  0 ;  hence, 
cos  h  =  0;  .  • .  h  =  90°,  or  PB  =  90°. 

sin  (90°— B}  =  tan  p  tan  (90°— A),  which  reduces  to 
cos  B  =  tan  p  cot  A. 

Since  h  =  90°,  cot  A  =  0;  .  • .  tan  p  cot  h  ^=  0; 
.-.  cos  B  =  0;  .-.  B  =  90°. 


PH '  =  180°  -  P#  =  90° ;     .  • .  PR'  =  PH. 
PB'  =  180°  —  PB  =  90°  ;     .  • .  PB'  =  PB. 


EIGHT:  .TRIANGLES.  117 

Hence,  if  P  <  90°  and  p  =  P,  b  =  90°,  A  *a  90°, 
J3--900,  the  two  triangles. reduce,  to  the  bi-rectangular 
triangle  PHB,  and  there  is  but  one  solution. 

3.  When  P  <  90°  and  p  >  P. 

As  before,  we  have  sin  b  =  tan  p  cot  P. 

Since  p  and  P  are  of  the  same  species,  p  <  90°. 

Then,  if  p  >  P,  tan  p  >  tan  P;  but  tan  P  cot  P  =  1; 
.  • .  tan  p  cot  P  ]>  1 ;  .  • .  sin  b  ]>  1,  which  is  impossible. 

Hence,  if  P  <  90°  and  p  >  P,  no  solution  is  possible. 

4.  When  P  >  90°  and  j>  >  P. 

K    /V  V 

We  have  sin  b  —  tan  p  cot  P, 
as  before,  tan  p  and  cot  P  are 
both  negative,  and  tan  p  <  tan  P,  numerically;  but 
tan  P  cot  P  =  1  ;  .  • .  tan  p  cot  P  <  1 ;  hence, 
sin  b  <  1 ;  .  • .  b  <  90°,  or  b  >  90° ;  hence,  b  may 
be  either  of  the  supplementary  arcs  PH  or  PH'  which 
have  the  common  sine  equal  to  tan  p  cot  P. 

If  6  <  90°,  since  p  >  90°,  h  >  90° ;  if  6  >  90°, 
since  p  >  90°,  h  <  90°. 

Hence,  if  P*>  90°  and  p  >  P,  either  triangle,  PHB 
or  PH'B'  will  satisfy  the  conditions,  and  there  will  be 
two  solutions. 

5.  When  P  >  90°  and  p  =  P.  x^~~T"\ 

P\  /r 

We  have  sin  b  =  tan  »  cot  P,  as        \.       /    jx^ 

^""'•^     /      — ""x^ 

before.  H 

.  • .     sin  b  =  tan  P  cot  -P=  1 ;     .  • .    b  =  90°. 
.  • .     cos  b  —  0;    . ' .  cos  h  =  cos  b  cos  jo  —  0 ;    .  • .  k  =  90°. 

=  0;    .-.  5  =  90°. 


118  TRIGONOMETRY. 

Hence,  if  P  >  90°  and  p  =  P,  b  =  90°,  h  =  90°, 
B  =  90°,  the  two  triangles  reduce  to  the  bi-rectangular 
PHB,  and  there  is  but  one  solution. 

6.  When  P  >  90°  and  p  <  P. 

As  before,  we  have  sin  b  =  tan  p  cot  P. 

Since  p  and  P  are  of  the  same  species,  and  since 
P  >  90°,  p  >  90°  ;  hence,  tan  jo,  cot  P  are  both  nega- 
tive, and  tan  p  >  tan  P,  numerically;  but  since 
tan  P  cot  P  =  1,  tan  p  cot  P  >  1  ;  .  •  .  sin  b  >  1, 
which  is  impossible. 

Hence,  if  P  >  90°  and  p  <  P,  there  is  no  solution. 

7.  When  P==90°. 
sin  b         sin  6  . 


==^Fp=  IT   =oo;  •'•  ?=.•• 

.  •  .     cos  jo  =  0  ;   .  •  .  cos  A,  =  cos  b  cos  7)  =  0  ;  .  *  .  h  =  90°. 
sin  b  =  tan  p  cot  P=  oo  x  0;   .  *  .  sin  6  is  indeterminate. 

sin  B  =  -  •=  -pr-  ;   .  •  .  sin  5  is  indeterminate. 
cos  p         0 

Hence,  if  P=90,  then  JD  =  90°,  h  =  90°,  6  and  B  are 
indeterminate  ;  the  triangle  is  bi-rectangular,  and  there 
is  an  infinite  number  of  solutions. 

Hence,  the  following  results  : 

(  p  <  P,     Two  solutions. 
P  <  90°  and     |  p  =  P,     One  solution. 
I  p  >  P,     No  solution. 

r  p  >  P,     Two  solutions. 
P  >  90°  and    <  JD  =  P,     One  solution. 
(  ^  <  P,     No  solution. 


RIGHT  TRIANGLES.  '11.9 

P  g=  90°,  x 

/i  =  90°,  f   Infinite  number 

ft  indeterminate,  f      of  solutions. 
5  indeterminate,  J 

By  a  comparison  of  these  results,  we  find, 

1.  If  jo  differs   more    from  90°  than  P,  there  will   be 
two  solutions. 

2.  If  p  —  P,  and  P  <  90°  or  P  >  90°,  there  will   be 
one  solution. 

3.  If  p  =  P  =  90°,  there  will  be  an  infinite  number 
of  solutions. 

4.  If  p  differs  less  from  90°  than  P,  there  will  be  no 
solution. 

130.   Remarks. 

1.  Napier's    principles    render   it    unnecessary  to   di- 
vide   the    subject    of    right-angled    spherical    triangles 
into  cases. 

2.  Two  parts  will  be  given,  and  three  required. 

3.  These  parts  or  their  complements  will  be  circular 
parts. 

4.  Take    the    two    given    parts,   if   they  are    circular 
parts,  otherwise  their  complements,  and  any  one   part 
required,  if  it  is  a  circular  part,  otherwise  its  comple- 
ment,  and    observe    which    is    the    middle    part,    and 
whether   the   other  parts  are  adjacent   to,  or  opposite, 
the    middle    part :    if    adjacent,   the    first    of   Napier's 
principles    will    give    the    formula;    if    opposite,    the 
second. 

5.  Introduce  R  and  apply  logarithms. 

6.  Apply  the  principles  which  determine  the  species 
of  the  required  part. 


120  '   TEIGONOMETR F. 


h  =  110°  30'. 
1.  Giv.1  5Qo45 


1.  To  fjnd  b. 

From  the  second  of  Napier's  principles,  we  have 
sin  (90°  —  K)  —  cos  b  cos  p,  or  cos  h  =  cos  b  cos  p. 
Finding  cos  b  and  introducing  7?,  we  have 

R  cos  A 

cos  b  —  — 

cos  p 

, ' .    log  cos  b  =  10  -{-  log  cos  A  —  log  cos  p. 

log  cos  h  (110°  3CX)  '==  9.54433  - 
log  cos  p  (  50°  45' )  ==  9.80120  + 

log  cos  b  =  9.74313  —.-.&=  123°  36'  31". 

Since  the  hypotenuse  is  greater  than  90°,  the  sides 
b  and  p  are  of  different  species  ;  but  p  <  90° ; 
.  • .  b  >  90°.  But  log  cos  b  corresponds  to  56°  23'  29", 
and  to  its  supplement  123°  36'  31"  which  must  be 
taken,  since  b  >  90°. 

The  species  of  b  can  also  be  determined  by  the 
form'ula, 

cos  h 


cos  b  — 


cos  p 


Since  h  >  90°,  cos  h  is  negative,  and  since  p  <  90°, 
cos  p  is  positive;  .'.  cos  b  is  negative;  . '.  b  >  90°. 
The  signs  of  the  functions  may  be  conveniently  indi- 
cated by  placing  the  signs  after  their  logarithms. 


RIGHT  TRIANGLES.  121 

2.  To  find  B. 

sin  (90°  —  B)  =  tan  p  tan  (90°  —  K), 
tan  p  cot  h 

.-.      COSJB=  -^- 

.  *  .     log  cos  5  =^=  log  tan  jp  -|-  log  cot  h  —  10. 

log  tan  p  (  50°  45')  =  10.08776  + 
log  cot  h  (110°  30')  £=   9.57274  ~ 

log  cos  B  --   9.66050-      ..-.  B=  117°  14'. 

Since  6  and  B  are  of  the  same  species,  and  since 
b  >  90°,  B  >  90°.  The  species  of  B  can  also  be  de- 
termined from  the  sign  of  cos  B. 

3.  To  find  P. 

sin  p  =  cos  (90°  —  h)  cos  (90°  —  P),  or  sin  p  =  sin  h  sin  P. 

n      P  sin  p 
.  •  .    gin  £*=  —  .  —  ,-  ; 
sin  h    ' 

.  •  .     log  sin  P  =  10  -J-  log  sin  j9  —  log  sin  A. 

log  sin  j9  (  50°  45')  ==  9.88896  + 
log  sin  h  (110°  30')  =  9.97159  -f- 

log  sin  P  -  9.91737  +  •'•  P=^°  &'  57". 

P  is  of  the  same  species  as  p,  and  since  p  <  90°, 
P  <  90°.  The  species  of  P  can  not  be  determined  by 
the  sign  of  sin  P,  since  the  sign  of  sin  P  is  plus 
from  0°  to  180°. 

:    67°33'27"' 

2.  Given  Re.     B=   67°  54'  47". 

=   99°  5735". 

=    67°  06'  44". 

3.  Given         = 


(     h         94°  05'      I  f: 

\  __  ^  ^/      \     Req.  1  B= 

1     p=  \P= 

r^       110^46'  26"!  (6=    67°  06'  44". 

{  h  =  l1^  f  ^  }     Req.  \p  =  155°  47'  05", 

I  P=  153^  58'  45". 


S.  N.  11. 


122  TRIGONOMETRY '. 


=.    990       ,       ,,  f  #  =    54  3  01'  15". 


. 

4.  Given  j  p  =  13?0  ^  2r  |     Req.  j  *=^  142°  09'  12". 

(  jo  =  155°  27'  55". 

r      ,       fi«0  1V       ,  fh=   75°  13'  01". 

5.  Given]  ~  "    Zf  ReqJ  *  =   67°  27'  01". 

lp  =   58°  25'  45". 


rP^-5°^0'l     • 
.  Giv.  ]      ^Lo-,,/  f 

H 


=  52°  34'  31"  or  127°  25'  29'. 
B  =  23°  03'  06"  or  156°  56'  54". 


8.  If  a  line  make  an  angle  of  40°  with  a  fixed  plane, 
and  a  plane  embracing  this  line  be  perpendicular  to 
the  fixed  plane,  how  many  degrees  from  its  first  posi- 
tion must  the  plane  embracing  the  line  revolve  about 
it  in  order  that  it  may  make  an  angle  of  45°  with,  the 
fixed  plane?  Ans.  67°  22'  44"  or  112°  37'  16". 


132.    Polar  Triangles. 

The  polar  triangle  of  a  given  triangle  is  the  triangle 
formed  by  the  intersection  of  three  arcs  of  great  circles 
described  about  the  vertices  of  the 
given  triangle  as  poles. 

If  one  triangle  is  the  polar  of  an- 
other, the  second  is  the  polar  of  the 
first. 

Thus,  if  A'B'C'  is  the  polar  of  the 
triangle  ABC,  then  ABC  is  the  polar  of  A'B'C'. 

Each  angle  in  one  of  two  polar  triangles  is  the  sup- 
plement of  the  side  lying  opposite  to  it  in  the  other; 


RIGHT  TRIANGLES.  123 

and  each  side   is  the  supplement  of   the  angle  lying 
opposite  to  it  in  the  other.     Thus, 

A  =  180°  —  a',  B  =  180°  —  &',  C  =  180°  —  c'. 
a  =  180°—  A',  6  =  180°—  B',  C  =  180°  —  C'. 
A'=  180°  —  a,  B'=  ISO0  —  b,  C'=  180°  —  c. 


Cor.—  If  a'  =  90°,  A  =  90°  ;    hence,  if  one  side  of  a 
triangle  is  90°,  one  angle  of  its  polar  triangle  is  90°. 


133.   Quadrantal  Triangles. 

A  quadrantal  triangle  is  a  triangle  one  side  of  which 
is  90°. 

By  the  corollary  of  the  last  article,  it  follows  that 
the  polar  of  a  quadrantal  triangle  is  a  right-angled 
triangle. 

A  quadrantal  triangle  is  solved  by  passing  to  its 
polar  triangle,  which  is  solved  as  a  right-angled  tri- 
angle, then  by  passing  back  to  the  quadrantal  triangle, 
which  is  the  polar  of  the  right-angled  triangle. 


134.   Examples. 

rh'=   90°.  ^  fH'=  69°  30'. 

1.  Given  {  F  =  129°  15'.         V  Req.  \B'  =   56°  23'  30". 
I  &'  =    62°  46'  01". )  I  p'  =  124°  14'  03". 

Passing  to  the  polar  triangle,  which  is  right-angled, 
we  have 

fH=   90°.  ^  f  h  =  110°  30'. 

Given  \  p  =   50°  45':         V    . '.   <    b  =  123°  36'  30". 
I B  =  117°  13'  59".  J  I  P  =    55°  45'  57". 


124  TRIGONOMETRY. 

Passing    back    to    the    quadrantal    triangle,    we    find 

f  a'=:    90°.  ^  (A'=    74°  26'. 

%  Given  <  c'  ==    99°  20'.         V  Req.K  C"  =  108°  05'  26'. 
(B'=   30°  12'  23".  j  I  &'  =    31°  29'  14'. 

OBLIQUE   TRIANGLES. 

135.    Proposition  I. 

The  sines  of  the  sides  of  a  spherical  triangle  are  propor- 
tional to  the  sines  of  their  opposite  angles. 

Let  ABC  be  a  spherical  tri- 
angle. From  C  draw  p,  the  arc 
of  a  great  circle  perpendicular 
to  the  opposite  side  or  to  the 
opposite  side  produced. 

In  the  first  case  we  have,  by  Napier's  principles, 
sin  p  =  cos  (90°—  a)  cos  (90°— 5)  &*  sin  a  sin  B. 
sin  p  =  cos  (90°—  ft)  cos  (90°— A)  =  sin  6  sin  A. 

. ' .     sin  a  sin  B  —  sin  b  sin  A. 
. ' .     sin  a  :  sin  b  :  :  sin  ^4  :  sin  B. 

In  the  second  case  we  have,  by 
Napier's  principles, 

sin  p  =  cos  (90°—  a)  cos  (90°— B')  ==    A 
sin  a  sin  B'=  sin  a  sin  5. 

sin  p  =  cos  (90°  —  ft)  cos  (90°  —  A)  =  sin  6  sin  A. 

.'.     sin  a  sin  B  =  sin  ft  sin  A 
.  • .     sin  a  :  sin  ft  : :  sin  A  :  sin  B. 


OBLIQ  UE  TRIA  NGL  ES. 


125 


In   like  manner  other  proportions  may  be  deduced, 
giving  the  group, 

(1)  sin  a  :  sin  b  ::  sin  A  :  sin  B. 

(2)  sin  a  :  sin  c  :  :  sin  A  :  sin  C. 

(3)  sin  b  :  sin  c  : :  sin  B  :  sin  C. 


136.   Proposition  II. 

The  co-sine  of  any  side  of  a  spherical  triangle  is  equal  to 
the  product  of  the  co-sines  of  the  other  sides,  plus  the  product 
of  their  sines  into  the  co-sine  of  their  included  angle. 

Let  ABC  be  a  spher- 
ical triangle,  and  0  the 
center  of  the  sphere. 

Let  CM  be  perpendic-  ° 
ular  to  the  plane  AOB. 
Draw  MD  and  ME,  re- 
spectively perpendicu- 
lar to  OB  and  OA,  and 
draw  CD  and  CE,  which  will  be  respectively  perpen- 
dicular to  OB  and  OA;  hence,  the  angle  OEM  —  A. 
and  CDM  ==  B.  Draw  EF  perpendicular  to  OB,  and 
MN  perpendicular  to  EF.  Each  of  the  angles  MEN  and 
EOF  is  the  complement  of  OEF;  .  • .  MEN=EOF=  c. 

OD  =  OF  -f-  NM. 

OD  =  cos  a. 

OF  =  OE  cos  EOF  =  cos  6  cos  c. 

NM=  EM  sin  MEN  =  sin  b  cos  A  sin  c. 

Substituting  the  values  of  OD,  OF,  and  NM,  we  have 
cos  a  —  cos  b  cos  c  -f-  sin  6  sin  c  cos  A. 


TRIGONOMETRY. 

In  like  manner  other  formulas  may  be  deduced,  giv- 
ing the  group, 

(1)  cos  a  =  cos  b  cos  c  -\-  sin  b  sin  c  cos  A. 

(2)  cos  b  ~  cos  a  cos  c  -f-  sin  a  sin  c  cos  J?. 

(3)  cos  c  =  cos  a  cos  6  -j-  sin  a  sin  b  cos  (7. 


137.    Proposition  III. 

The  co-sine  of  any  angle  of  a  spherical  triangle  is  equal  to 
the  product  of  the  sines  of  the  other  angles  into  the  co-sine 
of  their  included  side,  minus  the  product  of  the  co-sines  of 
these  angles. 

The  formulas  for  passing  to  the  polar  triangle  are, 

a  =  180°  —A',     b  =  180°  —  £',     c  =  180°  —  C'. 
A  =  180°  —  a',     B  =  180°  —  6',     C  =  180°  —  c'. 

Substituting  these  values  in  the  formulas  of  the 
preceding  article  and  reducing,  we  have 

—  cos  A'=  cos  B'  cos  C" — sin  B'  sin  C'  cos  a'. 
—  cos  B'=  cos  A'  cos  C' — sin  A'  sin  C"  cos  b'. 

—  cos  C'=  cos  -4'  cos  B' — sin  A'  sin  B'  cos  c'. 

Changing  the  signs  and  omitting  the  accents,  since 
the  formulas  are  true  for  any  triangle,  we  have 

(1)  cos  A  —  sin  B  sin  C  cos  a  —  cos  B  cos  C. 

(2)  cos  B  =  sin  A  sin  C  cos  b  —  cos  A  cos  C. 

(3)  cos  C  =  sin  A  sin  J5  cos  c  —  cos  -4  cos  B. 


138.    Proposition  IV. 

7Vi£  co-sine  of  one- half  of  any  angle  of  a  spherical  tri- 
angle is  equal  to  the  square  root  of  the  quotient  obtained  by 


OBLIQUE  TRIANGI.IW.  127 

dividiinj  tin'  ,s///r  <>f  true-half  th<  xmit  of  (!/<'  sides  into  tin- 
vine  of  o'lH'-hatf  the  .s/rwi  minus  the  M</<  o/^ax/Vr  ////•  timjlr, 
by  tJie  product  of  flic  xincx  of  tin-  aa'jarcnf  -svVfx. 

The  first  formula  of  article  130  gives 
cos  a  —  cos  6  cos  c 

COS   A  —  :  -  :  -  ;  — 

sin  b  sin  c 
Adding  1  to  botli   im-mlMTs,  we   have 

cos  a  -f  sin  6  sin  c  —  cos  b  cos  c 
1  4-  cos  A  =  —  :  —  7  —  r—  —  • 

sin  />  sin  c 

1  +  cos  A  =  2  cos2  £  A     Article  95,  (10). 
sin'  b  sin  c  —  cos  fr  cos  c  =    -  cos  (/>  H   c).     Art.  89,  (tf). 


. 
sin  ft  sm  c 

But  by  article  96,  (8),  we  have 

cos  a  —  cos  (ft  H-  c)  =  2  sin  J(a  -f  ft  -f-  c)  sin  J(6  +  c  —  a). 
Substituting  and  dividing  by  2,  we  have 

sin  K<*  4-  b  H-  c)*  sin  fr(ft  +  g  —  •  o) 

COS     •5^1  —  —  ;  -  ;  -  ;  — 

Sill   ft   Bill    c 

Let  8  —  a  -f  ft  -h  r,  then  will  £  s  ^  J  (a  -f  ft  +  <0> 
^8  —  a  —  J(ft  4-  c  —  a). 

Substituting  in  the  value  of  cos2  \A,  and  in  the 
similar  values  for  cos2  \E  and  cos2  JCT,  and  extracting 
the  square  root,  we  have 


(1)    cos  \A  = 


(2)    cos  \E  = 


's'  sn     iH 
sin  ft  sin  c 


s*  8n      s- 
sin  a  sin  c 


(3)    cos  JO  .-=Jri'n  forin  .  !         cj  _ 
^          sin  a  sin  ft 


128  TRIG  ONOMETR  Y. 

139.    Proposition  Y. 

The  sine  of  one-half  of  any  side  of  a  spherirnl  triangle  is 
equal  to  the  square  root  of  the,  quotient  obtained  In/  dividing 
minus  the  co-sine  of  one-half  the  sum  of  the  angles  into  the 
co-sine  of  one-half  the  sum  minus  the  angle  opposite  the  side, 
by  the  product  of  the  sines  of  the  adjacent  angles. 

Taking  the  formulas  of  the  last  article,  passing  to 
the  polar  triangle,  making  S  =  A'  -f  B'  -j-  C",  substitut- 
ing in  these  formulas,  reducing,  and  omitting  the  ac- 
cents, we  have 


(1)     sin  ^a  =  J  — CQS  £S  cos  QS  —  A) 
^  sin  B  sin  C 


(2)    sin  i  6  =  J  —  costs  cos  (££-£)  . 
*  sin  A  sin  (7 


(3)      sin   lf=      I-<X**8<X*QS- 

^  sin  A  sin  5 


140.   Proposition  VI. 

The  sine  of  one-half  of  any  angle  of  a  spherical  triangle  is 
equal  to  the  square  root  of  the  quotient  obtained  by  dividing 
the  sine  of  one-half  the  sum  of  the  sdd.es  minus  one  adjacent 
side  into  the  sine  of  one-half  the  sum  minus  the  other  adjacent 
side,  by  the  product  of  the  sines  of  the  adjacent  sides. 

cos  a  —  cos  6  cos  c 

cos  A  =  -    —. — r— T—       -  •    Article  136,  (1). 
sin  b  sin  c 

Subtracting  both  members  from   1,  we  have 

cos  b  cos  c  4-  sin  b  sin  c  —  cos  a 
1  —  cos  A  --  -  —. — i — r-r  -  • 

sin  6  sin  c 

1  —  cos  A  =  2  sin2  JA     Article  95,  (9). 


OBLIQUE  TRIANGLES.  129 

cos  b  cos  c  -}-  sin  b  sin  c  =  cos  (6  —  r).     Article  91,  (d).' 


.. 

sin  o  sin  c 

But  by  article  96,  (8),  we  have 
cos  (ft  —  6-)  —  cos  a  =  2  sin  J(a  -f  c  —  ft)  sin  J(a  +  6  —  c). 

Substituting  and  dividing  by  2,  we  have 

.    a  t  ,  _  sin  £(q  +  c  —  ft)  sin  |(a  +  ft  —  c) 

Sin      *^i   —  ;         j  -  ;  -  • 

sin  ft  sin  c 
But  J(a  -|-  c  —  ft)  =  £  s  —  ft  and  J(a  -f  ft  —  c)  =  %s  —  c. 

Substituting  in  the  value  of  sin2  J^4,  and  in  the 
similar  values  for  sin2  ^B  and  sin2  J(7,  and  extract- 
ing the  square  root,  we  have 


(1)     sin  \A  =      sm  ft  s  -  ft)  jmjjs--  c)  . 
^  sin  ft  sin  c 


/sin  (jg  —  a)  sin  (jg  —  c) 


(2)     sin  $B  = 

*  sin  a  sin  c 


(3)     sin  iC  =  J  sin  (i  «- a)  sin  (j  8 -6)  § 
^  sin  a  sin  ft 


141.   Proposition  VII. 

The  co-sine  of  one-half  of  any  side  of  a  spherical  triangle 
•is  equal  to  the  square  root  of  the  quotient  obtained  by  dividing 
the  co-sine  of  one-half  the  sum  of  the  angles  minus  one  adja- 
cent angle  into  the  co-sine  of  half  the  sum  minus  the  other 
adjacent  angle,  by  the  product  of  the  sines  of  the  adjacent 
angles. 


130  TRIGONOMETRY. 

Taking  the  formulas  of  the  last  article,  passing  to 
the  polar  triangle,  making  S  =  A'  -f  Br  -j-  C",  substitut- 
ing, reducing,  and  omitting  the  accents,  we  have 


sin  B  sin  C 


(1) 


(2)    ~»lh-       cos  QS-A)  cos  (jg= 
*  sin  A  sin  (7 


(3)    cos  \c  =     / 
v 


cos         -        cos 


sin  ^4  sin  B 


U2.   Proposition  VIII. 

The  tangent  of  one-half  of  any  angle  of  a  spherical  tri- 
angle is  equal  to  the  square  root  of  the  quotient  obtained  by 
dividing  the  sine  of  one-half  the  sum  of  the  sides  minus  one 
adjacent  side  into  the  sine  of  one-half  the  sum  minus  the  other 
adjacent  side,  by  the  sine  of  one-half  the  sum  of  the  sides  into 
the  sine  of  one-half  the  mm  minus  the  opposite  side. 

Dividing  (1),  (2),  (3),  article  140,  respectively,  by  (1), 
(2),  (3),  article  138,  we  have 


sin  Js  sin  (J*  —  a) 


(1)    tan  \A  =  Jsin(i«-6)8in(t*-g) 
*       sin     s  sin      *  —  a 


(2)    tan  JP  -  Jsin 
\        si 


sn      s  — 


sn     s  sn 


(3)     tan  iC  =     /  sin^—a)  sin  ($*  —  ft) 
*       sin  ^s  sin  (-Js —  c) 


143.    Proposition  IX. 

The  tangent  of  one-half  of  any  side  of  a  spherical  triangle 
is  equal  to  the  square  root  of  the  quotient  obtained  by  dividing 


OBLIQUE   TRIANGLES.  131 

minus  the  co-sine  of.  one-half  the  sum  of  the  angles  into 
the  co-sine  of  one-half  the  sum  minus  the  angle  opposite  the 
side,  by  the  co-sine  of  one-half  the  sum  of  the.  angles  minus 
one  adjacent  angle  into  the  co-sine  of  one-half  the  sum  minus 
the  other  adjacent  angle. 

Dividing  (1),  (2),  (3),  article  139,  respectively,  by  (1), 
(2),  (3),  article  141,  we  have 


~  COS     S  cos  ~ 


(1) 

^  cos  as  —  B)  cos  as  —  C) 
(2)    tan  Jfr==^ 


-  cos        cos 


—  A)  COB  as—c 

(3)     tan  $c  =  J~  -  co*  $S  cos  as  —  C) 

>  cos  as  —  A)  cos'  as  —  B) 


The  reciprocals  of  (1),  (2),  (3),  articles  142,  143,  will 
give  formulas  for  co-tangents,  which  may  be  written 
and  expressed  in  words. 


144.   Napier's  Analogies. 

Dividing  (1),  article  142,  by  (2),  we  have 

tan  J  A        sin  a  s  —  °) 
tan  J  B        sin  (J  -9  —  «) 

This,  as  a  proportion   taken   by  composition  and  di- 
vision, gives 

tan  J  A  +  tan  4  -B        sin  (£  s  —  6)  +  sin  (is  —  a) 


tan  \A  —  '  tan  \  B        sin  (-J  s  —  6)  —  sin  (J«  —  a) 

sin  %A        sin  \E 

tan  i  -4  -h  tan  ^  B  _  cos  J  ^4        cos 


tan  J  ^4  —  tan  \B~~  sin  ^^4  _  sin 
cos  J  A       cos 


132  TRIG  ONOMETR  Y. 

Multiplying   both   terms   of   the   second  member  ly 
cos  \A  cos  J#, 

tan  \A  +  tan  \B  _  sin  \A  cos  \E  -f  cos  \A  sin  \E 
tan  J^4  —  tan  \E       sin  \A  cos  J5  —  cos  \A  sin  £# 

Reducing  the.  second  member  by  articles  89,  (a),  and 
91,  (c\ 

tan  \A  4-  tan  ^7?  _  sin  \(A  -f  £) 
tan  i^  —  tarT       ~"  sin      A—B    ' 


sin  (jg  —  6)  -f  sin  tys  —  a}  tan  ^g 

sin  (is  —  6)  -sin  Qs  —  a)  ~tan"K«  —  6)  ' 

sin     (^  +  B)  tan     c 


'•     sin  \{A  —  B}  ~  tan  ^(a  —  6) 
•;  •  .    (1)  sin  %(A-f-B)  :  sin|(^4—  B)  :  :  tan  %c  :  tan  |(a  —  />). 

The  reciprocal  of  (1)  X  (2),  article  142,  gives 

1  _          sin  Js 

tan  \  A  tan  J  B  ~~  sin  (J  s  —  c) 

By  division  and  composition,  we  have 

1  —  tan  \  A  tan  ^5  __  sin  \s  —  sin  (\s  —  c) 
1  -f  tan  %A  tan  J  J5        sin  -|s  -j-  sin  (J«  —  c} 

Reducing  both  members  as  before,  we  have 

cos  %(A  +  B)  tan  \c 

cos  \{A  —  B)  ~^  tan  J  (a  +  6)  ' 

.  •  .    (2)  cos  JC^  +  B)  :  cos  tfA—E)  :  :  tan  Jr  :  tan  J(«  +  &)• 


Passing   from  (1)  and  (2)  to   the    polar   triangle,  we 
have 


(3)  sin  J(a  +  6)  :  sin  $(a—b)  :  :  cot  \G  :  tan  \(A— 

(4)  cos  \(a  +  6)  :  cos  Ma  —  6)  :  :  cot  \C  :  tan 


OBLIQUE  TRIANGLES.  133 

145.   Proposition. 

In  a  right-angled  spherical  triangle,  as  b  increases  from  0° 
to  90°,  from  90°  to  ISO3, /row  180°  to  270°,  and  from  270° 
to  360°,  if  p  <  90°,  h  increases  from  p  to  90°,  from  90°  to 
180°  —  p,  decreases  from  180"  —  p  to  90 D,  <md  ;>om  90° 
to  IV  (/  P  >  90°,  ^  decreases  from  p  to  90°,  /rom  90°  to 
180°  — jo,  increases  from  180^  —  p  to  90°,  and  /rom  90° 
to  p;  if  p  =  90°,  h  =  90°  /or  aM  values  of  b. 

1.    p  <  90°;     .•'.  cos  p  is  positive, 
cos  h  =  cos  ft  cos  p. 

If  6  —  0,  cos  6  =  1;     therefore, 
cos  h  =  cos  £>/    .  • .  h  =  p. 

As  b  increases  from  0°  to  90°,  cos  6  is  positive, 
and  diminishes  from  1  to  0;  '  .'.  cos  h  is  positive, 
and  diminishes  from  cos  p  to  0;  .  •.  h  increases  from 
p  to  90°. 

As  b  increases  from  90°  to  180°,  cos  b  is  negative, 
and  increases  numerically  from  0  to  —  1 ;  .  • .  cos  h  is 
negative,  and  increases  numerically  from  0  to  —  cos  p; 
.  • .  h  increases  from  90°  to  180°  —  jo,  and  the  triangle 
becomes  the  lune  HH'. 

As  6  increases  from  180°  to  270°,  cos  b  is  negative, 
and  decreases  numerically  from  ---  1  to  0;  .*.  cos  h  is 
negative,  and  decreases  numerically  from  —  cos  p  to  0; 
.'.  h  decreases  from  180°  —  p  to  90°. 

As  b  increases  from  270°  to  360°,  cos  b  is  positive, 
and  increases  from  0  to  1 ;  .  ' .  cos  h  is  positive,  and 
increases  from  0  to  cos  p;  ..'.  h  decreases  from  90° 
to  p,  and  the  triangle  becomes  the  hemisphere. 


134  TRIGONOMETRY. 

2.    p  >  90° ;     .  • .  cos  p  is  negative. 

TJ 

cos  h  =  cos  6  cos  p. 

-=—     =!• 

If  6  =  0,  cos  6  =  1;     therefore, 
cos  h  =  cos  p ;     .' .  h  =  p. 

As  6  increases  from  0°  to  90°,  cos  6  is  positive,  and 
decreases  from  1  to  0 ;  .  * .  cos  h  is  negative,  and  de- 
creases numerically  from  cos  p  to  0 ;  .  • .  h  decreases 
from  p  to  90°. 

As  6  increases  from  90°  to  180°,  cos  6  is  negative, 
and  increases  numerically  from  0  to  —  1 ;  .  • .  cos  h  is 
positive,  and  increases  from  0  to  —  cos  p;  .  • .  h  de- 
creases from  90^  to  180°  —  p,  and  the  triangle  becomes 
the  lune  HH'. 

As  6  increases  from  180^  to  270°,  cos  6  is  negative, 
and  decreases  numerically  from  —  1  to  0;  .  • .  cos  h  is 
positive,  and  decreases  from  —  cos  p  to  0;  .  •.  h  in- 
creases from  180°  —  p  to  90°. 

As  6  increases  from  270°  to  360°,. cos  6  is  positive, 
and  increases  from  0  to  1;  .  *.  cos  h  is  negative,  and 
increases  numerically  from  0  to  cos  p;  .'.  h  increases 
from  90°  to  p,  and  the  triangle  becomes  the  hemi- 
sphere. 

3.    p  =  90° ;     .  • .  cos  p  =  0. 
.  • .  cos  h  =  cos  6  cos  p  =  0 ;     .  • .  h  =  90°. 

Cor. — Since  B  and  6  are  of  the  same  species,  B  may 
be  substituted  for  6  in  the  preceding  proposition. 

In  the  application  of  these  principles  to  the  discus- 
sion of  Case  I,  in  which  two  sides  and  an  angle  oppo- 
site one  of  them  are  given,  a  corresponds  to  A,  and 
HB  to  6. 


OBLIQUE  TRIANGLES.  135 

146.   Case  I. 

Given  two  sides  of  a  spherical  triangle,  and  the  angle  oppo- 
site one  of  them;  required  the  remaining  parts. 

Let  a  and  b  be  the  given   sides 
and  A  the  given  angle. 

A<         P 


sin  p  —  sin  b  sin  A. 

1.     a  =  p. 

B  coincides  with  H,  and  the  triangle  ABC  becomes 
the  right  triangle  AHC. 

2.     a  <  90°  and  a  >  p. 

By  the  last  proposition  the  point  B  lies  in  the  first 
or  fourth   quadrant,   estimated  from  H. 

3.     a  =  90°. 
HB  =  90°  or  270°,  and  HCB  =  90°  or  270°. 

4.     a  >  90°  and  .«  <  180°  —  p. 
B  lies  in  the  second  or  third  quadrant  from  H. 

5.  a  =  180°  —  p. 

HB  ==  180°,  and  ABC  =  AHC  -f  J  the  hemisphere. 

6.  a  ==  180°  —  b. 


Q=  #.4'  or  360°  —  HA',   and  then  the   first   tri- 
angle becomes  the  lune  AA'. 

7.     a  =  6. 


HB  ^  AH  or  360°  —  ^#,  and  the  second  triangle 
becomes  the  hemisphere. 


136  TRIG  ONOMETR  Y. 

8.     a  <  p  or  a  >  180°  —  p. 

The  triangle  is  impossible,  since  p  is  the  least,  and 
1 80°  — -  p  is  the  greatest  value  of  a. 

II.     yl  >  905;     .'.   p  >  90°.  -£ 

.         ,        .  A<(^  W   «  \j^ 

sin  p  =  sin  6  sin  A.  N.  7  I  x- 

1.     a=p. 
B  coincides  with  H,  and  ABC  becomes  AHC. 

2.     a  >  90°  and  a  <  p. 
B  lies  in  the  first  or  fourth  quadrant  from  H. 

3.    a  =  90°. 
#£  =  90°  or  270°,  and  HOB  =  90°  or  270°. 

4.     a  <  90°  and  a  >  180°  —  p. 
B  lies  in  the  second  or  third  quadrant  from  H. 

5.  a  =  180°  —  p. 

HB  =  180°,  and  ABC  =  AHC  +  £  the  hemisphere. 

6.  a  ==  180°  —  6. 

HB  =  #4'  or  360°  —  /L4',  and  the  first  triangle  be- 
comes the  lune  A  A. 

7.     0  =  6. 

HB  -=  ,4#  or  360°  —  AH]  and   the   second   triangle 
becomes  the  hemisphere. 

8.     a  >  p  or  a  <  180°  —  ^. 

The   triangle   is   impossible,  since  p  is  the   greatest, 
and  180°  —  p  is  the  least  value  of  a. 

III.     A  ==  90°. 

The    triangle    is    right-angled,    and   is    solved   as    in 
article  131. 


OBLIQUE  TRIANGLES.  137 

147.   Examples. 

(  a  =  60D  20'.  ^  f  B. 

1.  Given  I  b  =  80°  35'.  V  Req.  1  C. 

A  <  90°;    .-.  p  <  90°. 
sin  p  =  sin  b  sin  A,     .  • .  p  =  37°  48'  26". 

Since   a  >  p  and  a  <   180°  --  p,   the    triangle    is 
possible. 

Since  a  <  6  and  a  <  180°  —  6,  B  lies  between  If 
and  A  or  H  and  .4'. 

sin  p  =  sin  a  sin  J5,  .  - .  B  =  44°  52'  05". 

cos  #<?£  =  tan  p  cot  a,  .  • .  HCB  =  63°  46'  18". 

cos  a  =±  cos  p  cos  #£,  .  • .  HB  =  51°  12'  41". 

cos  ACH  =  tan  p  cot  ft,  .  • .  ^ICTT  ==  82°  36'  25". 

cos  b  =  cos  p  cos  AH,  .'.AH  =  78°  02'  54". 

C  =  ACH  ±  HCB  =  146°  22'  43"  or  18°  50'  07". 

c  =  AH  ±  HB  =  129°  15'  35"  or  26°  50'  13". 

In  ACB,  ABC  =  180°  —  HBC  =  135°  07''  55". 

We  can  also  find  B  from  the  proportion, 
sin  a  :  sin  b  : :  sin  A  :  sin  B. 

C  and  c  can  be  found  from  the  proportions, 
sin  i(6  +  a)  :  sin  J  (6  —  a)  : :  cot  JC  :  tan  %(B  —  A). 
sin  ^4  :  sin  C  : :  sin  a  :  sin  c. 

2.  Given.  Required. 

f  a  =  63°  5<y.  ^|  f  5=   59°  16'  00"  or  120°  44'  00". 

<    b  =  80°  19'.   V  <   C  =  131°  29'  42"  or  24°  37'  30". 

1  A  =  51°  3(y.  j  I  ?  =  120°  47'  50"  or  28°  32'  44". 
S.  N.  12. 


138  TRIG  ONOMETR  Y. 


3. 

Given. 

Required. 

a  = 

=    75° 

38'.  -) 

rB=     65° 

28' 

or 

114 

0 

32'. 

b  =  104° 

22'.    V 

l\q=  180° 

or 

57° 

03' 

32". 

A  = 

=    65° 

28'.    ) 

(  c  =  180° 

or 

63 

3 

20' 

18". 

4. 

Given. 

R 

equ'i 

[red. 

a 

-.-  - 

99°  407 

48".  ^ 

(B  =  114° 

26' 

50" 

or 

65 

0 

33' 

10". 

b 

= 

64°  23' 

15".  V 

1  C  =  236° 

51' 

27" 

or 

97 

0 

27' 

13". 

A 

= 

95°  38' 

W.) 

U  =  236° 

or 

51" 

or 

100 

0 

49' 

49". 

5. 

Given. 

R 

equ\ 

a 

s= 

100°. 

) 

(B=    50° 

47' 

41" 

or 

129 

0 

12' 

19". 

b 

= 

85°. 

\ 

1  (7=186° 

05' 

16" 

or 

342 

o 

03' 

12". 

A 

== 

50°. 

} 

1  c  ==  187° 

50' 

09" 

or 

336 

o 

39' 

45". 

6.  If  A  <  90°,  what  is  the   relation  of  a  to  p,  or  to 
180°  —  pj  when  there  is  no  solution  ? 

7.  If  A  >  90°,  what   is  the  relation  of  a  to  p,  or  to 
180°  —  _p,  when  there  is  no  solution  ? 

148.   Proposition. 

In  a  right-angled  spherical  triangle,  as  B  increases  from 
0°  to  90°,  from  90°  to  180°,  from  180°  to  270°,  and  from 
270°  to  360°;  if  p  <  90°,  P  decreases  from  90°  to  p,  in- 
creases from,  p  to  90 D,  increases  from  90°  to  180°  —  p,  and 
decreases  from  180°  —  p  to  90°;  if  p  >  90°,  P  increases 
from  90°  to  p,  decreases  from  p  to  90°,  decreases  from  90°  to 
180°  —  p,  and  increases  from  180°  —  p  to  90°;  ifp  =  9Q°, 
P=.  90°,  for  all  values  of  B. 

V~"    ~~\H' 

*/v .       /H 

1.    jo  <  90°;     . •.   cos  p  is  positive.       /        ^^      / 

H(^_^XP 
cos  P   =  cos  j9  sin  J5. 

If  B  =0°,  sin  B  =  0;    .  • .  cos  P  =  0;    .-.P=90°. 


OBLIQUE  TRIANGLES.  139 

As  B  increases  from  0°  to  90°,  sin  B  is  positive, 
and  increases  from  0  to  1 ;  . '.  cos  P  is  positive,  and 
increases  from  0  to  cos  p;  .  • .  P  decreases  from  90° 
to  p. 

As  B  increases  from  90°  to  180°,  sin  B  is  positive, 
and  decreases  from  1  to  0;  . ',  cos  P  is  positive,  and 
decreases  from  cos  p  to  0 ;  .  • .  P  increases  from  p  to 
90°,  and  the  triangle  becomes  the  lime  HH'. 

As  B  increases  from  180°  to  270°,  sin  B  is  negative, 
and  increases  numerically  from  0  to  —  1 ;  .  • .  cos  P  is 
negative,  and  increases  numerically  from  0  to  —  cos  p; 
.  • .  P  increases  from  90°  to  180°  —  p. 

As  B  increases  from  270°  to  360°  sin  B  is  negative, 
and  decreases  numerically  from  —  1  to  0 ;  .  • .  cos  P  is 
negative,  and  decreases  numerically  from  —  cos  p  to  0; 
.  • .  P  decreases  from  180°  —  p  to  90°,  and  the  triangle 
becomes  the  hemisphere. 

2.    p  >  90° ;    .  • .  cos  p  is  negative. 

cos  P  =  cos  p  sin  B. 
If  B  =  0°,  sin  B  =  0',    .'.cosP=0;    .-.P=W°. 

As  B  increases  from  0°  to  90°,  sin  B  is  positive,  and 
increases  from  0  to  1 ;  .  • .  cos  P  is  negative,  and  in- 
creases numerically  from  0  to  cos  p;  .  • .  P  increases 
from  90°  to  p. 

As  B  increases  from  90°  to  180°,  sin  B  is  positive, 
and  decreases  from  1  to  0 ;  .  • .  cos  P  is  negative,  and 
decreases  numerically  from  cos  p  to  0 ;  .  • .  P  decreases 
from  p  to  90°,  and  the  triangle  becomes  the  lune. 

As  B  increases  from  180°  to  270°,  sin  B  is  negative, 
and  increases  numerically  from  0  to  —  1 ;  .  • .  cos  P  is 
positive,  and  increases  from  0  to  --  cos  p;  .'.  P  de- 
creases from  90°  to  180°  —  p. 


140  TRIGONOMETRY. 

As  B  increases  from  270°  to  360°,  sin  B  is  negative, 
and  decreases  numerically  from  —  1  to  0 ;  .  • .  cos  P  is 
positive,  and  decreases  numerically  from  —  cos  p  to  0; 
.'..P  increases  from  180°  --  p  to  90°,  and  the  triangle 
becomes  the  hemisphere. 

3.    p  =  90°;     .'•',  cos  p  =  0. 

. -.  cos  P  =  cos  p  sin  B  =  0;     .  • .  P  —  90°. 

Cor. —  Since  b  and  B  are  of  the  same  species,  b  may 
be  substituted  for  B  in  the  preceding  proposition. 

149.   Case  II. 

Given  two  angles  of  a  spherical  triangle  and  the  side 
opposite  one  of  them;  required  the  remaining  parts. 

Let  A  and  B  be  the  given  angles, 
and  b  the  given  side.  3- -£ 

A\    /Iy> 

I.     A  <  90°;     .  • :  p  <  90°.  ^^Jzffi 

sin  p  —  sin  b  sin  A. 

1.     B  >  p  and  B  <  90°. 

By  the  last  proposition,  the  point  B  lies  in  the  first 
or  second  quadrant  estimated  from  H  as  origin. 

2.     B  =  p. 
The  angle  HCB  =  90°,  and  the  arc  HB  ==  90°. 

3.     5  <  180°  —  p,  and  B  >  90°. 
5  lies  in  the  third  or  fourth  quadrant  from  H. 

4.     B  ==  180°  —  p. 
The  angle  HCB  =  270°,  and  the  arc  HB  =  270°. 


OBLIQUE  TRIANGLES.  141 

5.     B  =  90°. 

HB  =  0°,  180°,  or  360°,  and  the  triangle  becomes 
ACH,  ACH  -f-  |  of  a  hemisphere,  or  a  hemisphere  -f 
ACH. 

6.     B  =-  A. 

B  lies  in  the  first  or  second  quadrant  from  H,  and 
one  of  the  triangles  becomes  the  lune  A  A'. 

7.     B  =  180°-—A. 

B  lies  in  the  third  or  fourth  quadrant  from  H,  and 
one  of  the  triangles  becomes  the  hemisphere. 

8.     B  <  p  or  B  >  180°  —  p. 

The  triangle  is  impossible,  since  p  is  the  least,  and 
180°  —  p  is  the  greatest  value  of  B. 

II.     A  >  90°  ;     .  • .  p  >  90°. 
sin  p  =  sin  b  sin  A. 

1.     B  <  p  and  B  >  90°. 
jB  lies  in  the  first  or  second  quadrant  from  H. 

2.     B=p. 

The  angle  HCB  =  90°,  and  the  arc  HB  =  90°. 

3.     B  >  180°  —  p  and  B  <  90°. 
B  lies  in  the  third  or  fourth  quadrant  from  H. 

4.     B  =  180°  —  ;>. 
The  angle  tfC£  =  270°,  and  the  arc  HB  =  270°. 

5.     B  =  90°. 

HB  =  0°,  180°,  or  360°,  and  the  triangle  becomes 
ACH,  ACH  -f  £  of  a  hemisphere,  or  a  hemisphere  -f- 


142  TRIGONOMETRY. 

6.     B  =  A. 

B  lies  in  the  first  or  second  quadrant  from  H,  and 
one  of  the  triangles  becomes  the  lune  AA'. 

7.    B  =  180°  —  A. 

B  lies  in  the  third  or  fourth  quadrant  from  H,  and 
one  of  the  triangles  becomes  the  hemisphere. 

8.     B  >  p  or  B  <  180°  —  p. 

The  triangle   is   impossible,  since  p  is  the  greatest, 
and  180°  —  p  is  the  least  value  of  B. 

III.     A  =  90°. 

The   triangle    is    right-angled,   and    is    solved   as    in 
article  131. 


150.   Examples. 

(  A  =  75°  30'.  ^  r  a. 

1.  Giv.  <  B  =  80°  40'.  V   Req.  <  C. 

I  6  =  70°  50'.  J  I  c.      A 

A  <  90° ;     .  • .  p  <  90°. 

sin  p  =  sin  6  sin  A]    .  • .  p  =  66°  07'  56". 

Since   B  >  j?  and  <  180°  —  £>,  the   triangle   is  pos- 
sible. 

Since  B  <  90°  and  >  p,  B  lies  in  the  first  or  second 
quadrant  from  H. 

(    67°  56'. 
sin  p  =  sin  a  sin  J5,     .  • .  a  =  <  ^o  94' 

The  second  value  of  a,  the  supplement  of  the  first, 
is  taken  when  B  lies  in  the  second  quadrant  from  H. 


OBLIQUE  TRIANGLES. 


143 


f   23°  37'  44" 
cos  B       =  cosp  sin  HCB,   .  • .  HCB  =  <j  1560  / 


21°  48'  19". 
158°  11'  41". 


\  156°  22'  16". 
sin  HB   =  tan  p  cot  5,        , ; .  J7J3    =  | 

cos  ACH=-  tan  p  cot  6,          . ' .  ACH  =       38°  13'  36''. 
cos  6  =  cos  p  cos  v4#,          .-.  AH    =      35°  46'. 
C  =  ylO^/  +  JEfO5  ='61°  51'  20''  or  194°  35'  52". 
c  =  AH    +  HB      -  57°  34'  19"  or  193°  57'  41". 

We  can   find  a,  r,  and  C  from   the   proportions, 

sin  B  :  sin  A  :  :  s4n  ft  :  sin  a. 

sin  -J  (B  -}-  ^4)  :  sin  J (B  —  ^4)  : :  tan  J r  :  tan  \  (b  —  a). 
sin  b  :  sin  c  :  :  sin  J5  :  sin  C. 


2.  GrMtetl. 

^  =  :    33°  15'. 
B  =    31°  34' 38". 
b  =    70°  10'  30". 

3.  Given. 

A  =  132°  16'. 
B  =  139°  44'. 
b  =  127°  30'. 

4.  Given. 

A  =  48°  50'. 
5=131°  10'. 

b=    75°  48'. 


Required. 

a=  80°  03' 25"  or  99°  56' 35". 
C  =  161°  24'  52"  or  173°  30'  52". 
c  =  145°  03'  13"  or  168°  18'  23". 

Required. 

a  =  65°  16'  30"  or  114°  43'  30". 
C  =  165°  41'  46"  or  126°  40'  44". 
c  =  162°  2(T  55"  or  100°  07'  25". 


Required. 


a=    75°  48' 
C  =  360° 
c  =  360° 


or  104°  12'. 

or  328°  39'  28". 
or  317°  56'  42". 


Scholium. — In  the  two  preceding  cases  some  of  the 
parts  are  found  to  be  greater  than  180° ;  but  the  cor- 
responding triangles  conform  to  the  conditions  of  the 
problem,  and  are  therefore  true  solutions. 


144  TRIGONOMETRY. 

Parts  greater  than  180°  are  usually  excluded,  in 
which  case  the  principles  of  the  following  article  will 
aid  in  determining  the  species  of  the  parts. 

The  principles  established  in  Geometry  are  given 
without  demonstration. 


151.    Principles. 

1.  Each  part  of  a  spherical  triangle  is  less  than  180°. 

2.  The  greater  side  is  opposite  the  greater  angle,  and  con- 
versely. 

3.  Each  side  is  less  than  the  sum  of  the  other  sides. 

4.  The  sum  of  the  sides  is  less  than  360°. 

5.  The  sum   of  the  angles  is  greater  than  180°,  and  less 
than  540  \ 

6.  Each  angle  is  greater  than  the  difference  between    180° 
and  the  sum  of  the  other  angles. 

For,     A  +  B  +  C>  180°.     Principle  5. 
.-.     A  >  180°-  (B  -j-<7). 

The  last  formula  is  always  algebraically  true;  but  in 
case  B  +  C>  180°,  it  might  be  doubted  whether  it  is 
numerically  true. 

Passing  to  the  polar  triangle,  we  have,  by  principle  3, 

a'  <  V  +  c'. 

or     180°  —  A  <  180°  —  B  +  180°  —  C. 
or  -A  <  180°  —  (B  +  C). 

.-.     A  >  B  +  C—  ISO0. 

7.  A  side  differing   more  from  90°  than  another  side  is 
of  the  same  species  as  its  opposite  angle. 


OBLIQUE  TRIANGLES.  145 

By  article  136,  we  have 

cos  a  =  cos  b  cos  c  -f-  sin  6  sin  c  cos  A. 
cos  a  —  cos  b  cos  c 


cos  A  = 


sn      sn  c 


But  sin  6  sin  c  is  positive,  since  6  and  c  are  each 
less  than  180°' 

If  a  differs^  more  from  90°  than  ft  or  c,  then  we  shall 
have 

cos  a  >  cos  6,  or  cos  a  >  cos  c,  numerically ; 

and  since  neither  cos  b  nor  cos  c  exceeds  1,  we  have 
cos  a  >  cos  ft  cos  c. 

. ' .  cos  A  and  cos  a  have  the  same  sign,  .  • .  A  and  a 
are  of  the  same  species. 

8.  An  angle  differing  more  from  90°  than  another  angle 
is  of  the  same  species  as  its  opposite  side. 

By  article  137,  we  have 

cos  A  —  sin  R  sin  C  cos  a  —  cos  B  cos  C. 
cos  A  -{-  cos  B  cos  C 

.  '  .       COS   a  —   -       — - — — : — 

sin  B  sin  C 

If  ^4  differs  more  from  90°  than  B  or  C,  then,  as 
before,  cos  A  and  cos  a  have  the  same  sign,  or  A  and 
a  are  of  the  same  species. 

9.  Two  sides,  at  least,  are  of  the  same  species  as  their  oppo- 
site angles,  and  conversely. 

If  each  of  two  sides  differs  more  from  90°  than  the 
remaining  side,  they  will  be  of  the  same  species  as 
their  opposite  angles,  as  is  evident  from  principle  ,7, 

If  the  triangle  is  isosceles,  and  the  equal  sides  less 
than  90°,  the  perpendicular  from  the  vertex  to  the 
third  side  will  be  less  than  90°,  since  one-half  the 
S.  N.  13. 


146  TRIGONOMETRY. 

third  side  is  less  than  90°,  and  the  angles  opposite  this 
perpendicular  will  be  less  than  90°,  article  129,  or  of 
the  same  species  as  their  opposite  sides. 

If  the  equal  sides  are  greater  than  90°,  the  perpen- 
dicular will  be  greater  than  90°,  since  one-half  the 
third  side  is  less  than  90°,  and  the  angles  opposite  the 
perpendicular  will  be  greater  than  90°,  article  129,  or 
of  the  same  species  as  their  opposite  sides. 

If  one  side  exceeds  90°  by  as  much  as  90°  exceeds 
another  side,  and  the  third  side  is  greater  or  less  than 
each  of  the  other  sides,  this  third  side  is  of  the  same 
species  as  its  opposite  angle  by  principle  7. 

If  the  greater  of  the  two  sides  is  of  the  same  species 
as  its  opposite  angle,  then  we  shall  have  two  sides  of 
the  same  species  as  their  opposite  angles. 

If  the  greater  of  the  two  sides  is  not  of  the  same 
species  as  its  opposite  angle,  this  angle  will  be  of  the 
same  species  as  the  other  side,  or  less  than  90°  ;  but 
the  angle  opposite  this  other  side  is  less  than  the  angle 
opposite  the  greater  side,  and  hence  less  than  90°,  or 
of  the  same  species  as  its  opposite  side,  and  again  we 
have  two  sides  of  the  same  species  as  their  opposite 
angles. 

10.  The  sum  of  two  sides  is  greater  than,  equal  to,  or  less 
than,  180°,  according  as  the  sum  of  their  opposite  angles 
is  greater  than,  equal  to,  or  less  than,  180°. 


tan  Ka  +  &)  cos  %(A-\-B)  —  tan  Jc  cos  %(A—E).  Art.  144. 

But  c  <  180°,    .-.  Jc  <  90°,  tan  \c  >  0, 
and  A—B  <  180°,    .  -  .  %(A—B)  <  90°,  cos  %(A—E)  >  0. 
.  •  .    tan  \c  cos  $(A—B)  >  0,  tan  |(«+&)  cos  $(A+  E)  >  0. 
.  •  .    tan  J(a  -f  &)  and  cos  %(A  -f  B)  have  like  signs. 


OBLIQUE  TRIANGLES.  147 

If  %(A  +  B)  >,  =  or  <  90°,  J(a-f 6)  >,  =  or  <  90°. 
If  4  f  B  >,  =  or  <  180°,  a  +  6  >,  =  or  <  180°. 


152.    Case  III. 

Given  two  sides  and,  the  included  angle  of  a  spherical 
triangle;  required  the  remaining  parts. 

(a  =85°  W.\ 

1.  Given  <^  b  =  65°  40'.  V   Req.  {  B. 
I  C=  95°  50'.  J 

We  have,  article  144, 
cos  i(a  -f-  6)  :  cos  £  (a  —  6)  :  :  cot  \C  :  tan  \(A  -f  -B). 
sin  ^  (a-  -f  6)  :  sin  J  (a  —  b)  :  :  cot  -J  C  :  tan  %(A  —  B)* 

t$(A+'B)  =  74°  21'  49".  )  f  A  =  83°  29'  10''. 

\$(A  —  B)==   9°  07' 21".  j     *'•    \  B  =  65°  14'  28". 

We  also  have,  article  144, 

sin  %(A-{-E)  :  sin  %(A  —  £)  :  :  tan  J  c  :  tan  J  (a  —  b). 
.  • .  J  c  =  46°  43'  09",  '  .  • .  c  =  93°  26'  14". 

We  can  also  find  c  from  the  proportion, 

sin  A  :  sin  C  :  :  sin  a  :  sin  c. 

But  the  species  of  c  is  more  readily  determined  from 
the  proportion  employed;  for  if  we  take  the  supple- 
ment of  46°  43'  09",  then  c  would  be  greater  than  180°. 


Again,  all   the   known   terms   of  the    proportion   are 
positive;  hence,  tan  %c  is  positive,     .*.  ^c  <  90°. 

a  =  120°  30'  30".  ^  fA=  135°  05'  29". 

2.  Given  <•  b  =~70ft  20'  20".  V-  Req.  <  J5  =    50°  30'  09". 
50°-10<-10":  )  H  SP  69°  34'  58". 


(a  = 
I  Given  < •  b  = 

(c'=^ 


148 


TRIGONOMETRY. 


153.   Case  IV. 

Given  two  angles  and  the   included  side  of  a  spherical 
triangle;   required  the  remaining  parts. 


fA=   62°  54'. 
1.  Giv.  IB  =   48°  30'. 

I  c  =  114°  29'  58". 


c  :  tan  |  (a  -j-  b). 
c  :  tan  J  (a  —  6). 
f  a  ==  83°  12'  06". 

"  *\  C  *  \ 

We  also  have,  article  144, 

sin  \  (a  -J-  6)  :  sin  J  (a  —  6)  :  :  cot  j  (7  :  tan  £  (A  —  B). 
.  • .  |  C  =  62°  40',     .  • .  C  =  125°  20'. 


We  have,  article  144, 

-i 

cosion-*: 

>  :  cos  J  (4  —  B) 

:  :  tan 

sin  JC4-f  B: 

>  :  sin  1  (A  —  5) 

:'  :  tan 

(i(a  j_ 

6)  =  69°  55'  48". 

1 

(  A  =  126°  35'  02". 
2.  Given  <  B  =    61°  43'  58". 
I  c  =    57°  30'.     • 


a  =  115°  19'  57". 

b  =    82°  27'  59". 
C  =    48°  31'  38". 


154.   Case  V. 

Given  the  three   sides  of  a   spherical  triangle;    required 
the  angles.  0 

fa  ==  100°  49'  30".^|  f  A. 

1.  Giv.  <  b  =    99°  4(X  48".  V  Req.  <  B. 

U  =^    64°  23'  15". J  la 


By  article  138,  we  have 
cos  \A 


'sin  $s  sin 

^ 


—  a) 


sin  b  sin  r 


OBLIQUE  TRIANGLES.  149 

Introducing  R  and  applying  logarithms,  we  have 

log  cos  \A  =  \  [  log  sin  \  s  -}-  log  sin  (-J  s  —  a) 

-f-  a-  c.  log  sin  b  4-  a.  c.  log  sin  c]. 

.  • .  \A  =  48°  43'  14",       .  • .  A  =  97°  26'  28". 

,      f  B  =  95°  38'  00". 
In  like  manner  we  find     <  . 


a  =  85°  30'.          ^       .       f  A  =  83°  29'  08". 
2.  Given  <    b  =  65°  40'.  V   ReqJ   B  ==  65°  14'  20". 


r  a  = 
<  b  = 
I  c  =  93°  26'  18".  0  ==  95°  50'. 


155.   Case  VI. 

Given  the  three  angles  of  a,  spherical  triangle;  required 
the  sides. 

r4  =  119°  15'.}  fa. 

f.  Given  <£==    70°  39'.  V     Req .1  b. 

(  C  ==    48°  36'.  )  I  c. 

By  article  139,  we  have 


cos   >a  = 


sin  B  sin  C 
Introducing  R  and  applying  logarithms,  we  have 

log  cos  J  a  =  J  [log  cos  (-J&  —  B)  4-  log  cos  QS—C) 

4-  «..  6-.  log  sin  5  -j-  a.  c.  log  sin  C] . 

.-.  4 a  =  56°  11'  31",        .-.  a  =  112°  23' 02". 

/  b  =    89°  16'  54". 
In  like  manner  we  find    <     •         ^o  OQ/  QQ-/ 


r  ^  .  =  121°  36'  24".  ^  c  a  = 

X  B  =--   42°  15'  13".  >   ReqJ  ft  ^ 

I  C  =   34°  15'  03".  j  I  c  = 


^  121°  36'  24".  ^  f  a  =  76°  36'  00". 

2.  GivenX  B  ±i    42°  15'  13".  >  Req.  <  ft  bi±  50°  10'  40". 

40°  00'  20". 


150  MENSURATION. 

MENSURATION. 

156.  Definition  and  Classification. 

Mensuration  is  the  art  of  calculating  the  values  of 
geometrical  magnitudes. 

Mensuration  is  divided  into  two  branches  —  Mensu- 
ration of  surfaces  and  Mensuration  of  volumes. 

MENSURATION   OF   SURFACES. 

157.  Unit  of  Superficial  Measure. 

A  unit  of  superficial  measure  is  a  square  each  side 
of  which  is  a  linear  unit. 

Thus,  according  to  the  object  to  be  accomplished,  a 
square  inch,  a  square  foot,  a  square  yard,  an  acre,  etc., 
is  the  superficial  unit  taken. 

158.    Problem. 

To  find  the  area  of  a  rectangle. 

Let  k  denote  the  area,  b  the  base,  and  a  the  altitude 
of  a  rectangle. 

There  are  a  rows  of  b  superficial  units 
each. 

Since  there  are  b  superficial  units  in  one  row,  in  a  such 
rows  there  will  be  a  times  b  or  ab  superficial  units. 

.-.     (1)     k  =  ab. 

The  above  demonstration  applies  only  in  case  the 
base  and  altitude  are  commensurable,  or  have  a  com- 
mon .unit, ... 


SURFACES.  151 

If  the  base  and  altitude  are  incommensurable,  denote 
the  area  by  £',  the  base  by  6',  and  the  altitude  by  a'. 
Then,  since  by  Geometry  any  two  rectangles  are  to 
each  other  as  the  products  of  their  bases  and  alti- 
tudes, we  have 

I-  :  //  :  :   ab  :  a'V. 

But  k  =  ab,     .'.  k'  =  a'U. 

159.    Problem. 

To  find  the  area  of  a  parallelogram. 

1.  When   the   base   and   altitude   are    given. 

Let  k  denote  the  area,  b  the  base,  and 
a  the  altitude  of  a  parallelogram. 

Since  a  parallelogram   is  equal  to  a    [/ 
rectangle,  having   the   same   base  and 
altitude,   and   since  the  area  of   the  rectangle  is  equal 
to  the  product  of  its  base  and  altitude,  the  area  of  the 
parallelogram   is  equal   to  the  product  of  its  base  and 
altitude. 

.  • .     (1)     k  =  ab.          .  _______ 

A  7 

2.  When     two     sides     and     their 

included    angle    are    given. 


b 

a  =  c  sin  A.         .' .     (2)     k  =  be  sin  A. 


1GO.    Problem. 

To  find  the  area  of  a  triangle. 
1.  When   the   base   and   altitude   are   given. 

Since  a  triangle  is  one-half  the 
parallelogram  having  the  same  base 
and  altitude,  we  have  for  the  tri- 
angle, 

(1)     lt=\ab. 


152 


MENSURATION. 


2.  When  two  sides  and  their  included  angle  are  given. 

Since    a    triangle    is    one-half    the 
parallelogram,  having  an  equal  angle  c/ 

and  equal  adjacent  sides,  we  have  for         / 
the  triangle, 

(2)     k  ==  J  be  sin  A. 

3.  When  two  angles  and  a  side  are  given. 

The  third   angle  is   equal  to  180° 
minus  the  sum  of  the  given  angles. 

Let,  then,  the  angles  and  the  side       A 
b  be  given. 

By  the  last  case,  we  have 

k  =  %bc  sin  A. 

But     sin  B  :  sin  C  : :  b  :  c,      .  • .  . 

sin  B 

Substituting  this  value  of  e,  we  have 
b2  sin  A  sin  C 


b  sin  C 


(3)     *•= 


2  sin  B 


4.  When  two  sides  and  an  angle  opposite  one  of 
them  are  given.  IB 

Let  a  and  c  be   the   given   sides, 
and  A  the  given  angle. 

In  case  of  one  or  two  solutions  determined  by 
article  72,  find  the  value  or  values  of  C  and  B  from 
the  formulas, 

sin  c  =  C  Sm  A  ,    and  B  =±  180°  —  (A  +  C). 

Then,  by  (2),  we  have 

(4)  .  k  •=  £ac  sin  1?. 


SURFACES.  153 


5.  When  the  three  sides  are  given. 
Let  p  ---  the  perimeter  =  a  -f  b  -f  c. 
Then,  by  article  102,  we  have 

(5)     k  =  V~ 


6.  When  the  perimeter  and  angles  are  given. 

Let  p  be  the  perimeter,  and  A, 
and  C  the  angles. 

By  article  98,  (10),  (11),  (12),  A^ 

lp2  tan  \A  tan  \E  tan  \C=' 


...     (6)     jk  =  Jp2  tan  p  tan  ££  tan  JG 

7.  When   the   perimeter  and  radius  of  the  inscribed 
circle  are  given.  B 

Let  p  =  (i  +  6  +  <••>  and  y  be  the 
radius  of  the  inscribed  circle. 

ABC  =  ~ 


ABC=--k, 
...     jb=:  J(o  +  6  +  <?)  r;    but  a 
.'.     (7)     fc  = 


161.    Examples. 

1.  Find  the  area  of  a  triangle  whose  base   is  75  ft., 
and  altitude  is  24  ft.  Arts.  900  sq.  ft. 

2.  Two  sides  of  a  triangle  are  25  yds.  and  30  yds., 
respectively,  and  their  included  angle  is  50°  ;   required 
the  area.  Ans.  287.2665  sq.  yds. 


154  MENSURATION. 

3.  In   a   triangle,    b  =  100   ft.,    A  =  50°,    C—  60°: 
required  the  area.  Ans.  3529.9  sq.  ft. 

4.  In  a  triangle,  a  =  40  yds.,  c  =  50  yds.,  ^4  =  40°; 
required  the  area.  Ans.  998.18,  or  232.83  sq.  yds. 

5.  In   a  triangle,   a  =  12  ft.,  b  =  15  ft.,   c  =  17  ft.; 
required  k.  Ans.  87.75  sq.  ft. 

6.  In   a   triangle   the    perimeter   is   20   ft.,   and   the 
angles   are    50°,    60°,    and    70°,    respectively;    required 
the  area.  Ans.  18.85  sq.  ft. 

7.  In   a    triangle    the    perimeter   is    60   ft.,   and   the 
radius  of   the  inscribed   circle    is   5  ft.;    required  the 
area.  Ans.  150  sq.  ft. 


162.   Problem. 

To  find  the  area,  of  a  quadrilateral. 

1.  When  two  opposite  sides  and  the  perpendiculars 
to  these  sides  from  the  vertices  of  the  angles  at  the 
extremities  of  a  diagonal  are  given. 

Let  b  and  b'  be  two  opposite  sides, 
and  a  and  a'  the   perpendiculars  to       /i« 
these  sides  from  the  vertices  of  the    AL 
angles  D  and  B. 

ABCD  =  ABD       DCB. 


k  =    06 


Corollary  1.  —  If  b'  is  parallel  to  b,  the  quadrilateral 
becomes  a  trapezoid,  a'  =  a,  and  (1)  becomes 


.(2)     k  =  $a 


SURFACES.  155 

Corollary  2.  —  If   6'  —  6,    the    trapezoid    becomes    a 
parallelogram,  and  (2)  becomes 

(3)     k  =  ab. 

Corollary  3.  —  If  U  =  0,  the  trapezoid  becomes  a  tri- 
angle, and  (2)  becomes 

(4)     k  =  $db. 

2.  When   a  diagonal  and  the   perpendiculars  to  the 
diagonal  from  the   vertices  of  the  opposite  angles  are 
given.  B 

Let  d  denote  the  diagonal,  and  p 
and  p'  the  perpendiculars.  A 

ABCD  =  ABC  +  ADC. 
ABCD  =  k,        ABC  =  J  dp,        ADC  =  $  dp'. 

.-.     (5)    *  =  id(p+j/). 

3.  When  the  sides  and  a  diagonal  are  given. 

Let  the  areas  of  the  triangles  be  de- 
noted by  k'  and  k",  which  are  found 
by  article  160,  (5). 

.-.     (6)     Jk  =  ik'.-f  Ik". 

,     4.  When  the  sides  .and  one  angle  are  given. 

Draw  the  diagonal  opposite  the  given 
angle,  and  call  the  areas  of  the  tri- 
angles  k'  and  k".  A< 

In  one  triangle  we   have   two   sides 
and  their  included  angle,  from  which  we  find  the  area 
and  the  diagonal. 


156  MENSURATION. 

Then,  in  the  other  triangle,  we  have  the  three  sides, 
from  which  we  find  the  area. 

.-.     (7)    k  =  k'  +  k". 

5.  When  the  diagonals  and  their  included  angle  are 
given. 

Let  d  and  dr  denote  the  diagonals 
p  and  q,  r  and  s  their  segments, 
and  A  their  included  angle. 

The  angles  at  A  are  equal  or  sup- 
plementary; hence  their  sines  are  equal. 

BODE  ~=  BAC  -f  CAD  +  DAE  +  EAB. 

BCDE  =  fc,    BAC  =  %ps  sin  A,    CAD  =  %qs  sin  A. 

DAE  =  \qr  sin  A,     EAB  =  \pr  sin  A. 

.  •.     k  =  % (ps  -\-qs-\-qr-\-  pr)  sin  A. 
.But  ps  -f  9.9  -f  qr  +'pr  =  (p  -f  q).  (r  +  s)  =  <M. 
.'.     (8)     k  =  %dd'  sin  A. 


6.  When    the    angles    and    two    opposite    sides    are 
given.  0 

Let  a  =  EC,  and  b  =  AD. 

E  =  180°  —  (B  +  C). 

..-•• 

E 

The    angles    at    ^4    being 
supplementary,   their   sines    are    equal.      The    same    is 
true  of  the  angles  at  D. 

ABCD  =  BCE  —  ADE,    ABCD  =  k. 
a2  sin  B  sin  C  b2  sin  A  sin  D 


(9)    *  = 


2sin  E  2sin  £ 

a2  sin  £  sin  C       62  sin  A  sin 


2sin 


SURFACES.  157 

7.  When  three  sides  and  their   included  angles  are 
given. 

Let  a,  6,  and   c  be   the   given    sides, 
and  A  and  B  their  included  angles. 

ABCD  =  ABD  +  DBC. 
=  k,     ABD  =  \  ab  sin  A. 


Find   B'   and  rf,    B"  =  B  —  B',    DBC=%cd  sin  B". 
.  • .     (10)     k  —  ^  a&  sin  ^4  -f~  J  erf  sin  5". 

8.  When  the   sides  of  a  quadrilateral  inscribed  in  a 
circle  are  given. 

Let  a,  by  c,  rf  be  the  given  sides. 
ACBD=ACB  -\-ADB.  -^ 

A GBD  =  k,     ACB  =  \  ab  sin  C. 
ADB     =  \  erf  sin  D  --  \  erf  sin  (7, 
since  D  =  180°  —  (7. 

.  • .     fc  =  £  (a&  -j-  erf)  sin  C. 
~2  =  a2  -f-  62  —  2  a6  cos  (7,  article  97. 

==  c2  +  rf2  —  2  erf  cos  7)  =  e2  -f  rf2  -f  2  erf  cos  C. 
+  rf2  -f  2  erf  cos  (7  =  ft2  +  62  -  2  «6  cos  (7. 
a2  +  />2  —  e2  —  rf2 


cos      = 


2  (afe  -f-  erf) 


sin  C  =?»V  1.7-  cos2(7,     Let  s  =  a .  -f  6  -f-  e  -f  rf. 


2.|/(|8  — a)(|8— 6)(t8  — c)qg  — d) 

Sill   L/  —  ,  ; 

ab  -j-  co 

(11)    fc  =  i/(J«_0)  (Js  —  6)  (Js  — e)  (J«  —  rf). 


158  XfEXS  URA  TION. 

163.   Examples. 

1.  Two  opposite  sides  of  a  quadrilateral  are  35  rds. 
and   25    rds.,   and    the    perpendiculars    to    these    sides 
from  the  extremities  of  the  diagonal  are,  respectively, 
12  rds.  and  16  rds.;    required  the  area. 

Ans.  410  sq.  rds. 

2.  Find  the   area  of  a  trapezoid  whose  bases  are  15 
rds.  and  20  rds.,  and  whose  altitude  is  18  rds. 

Ans.  315  sq.  rds. 

3.  Two  adjacent  sides  of  a  parallelogram  are  30  rds. 
and  40  rds.,  and  their  included  angle  is  30° ;    required 
the  area.  Ans.  600  sq.  rds. 

4.  The  diagonal  of  a  quadrilateral  is  40  rds.,  and  the 
two  perpendiculars  to  the   diagonal   from  the   vertices 
of  the  opposite  angles  are  10  rds.  and  15  rds.,  respect- 
ively;   required  the  area.  Ans.  500  sq.  rds. 

5.  The  sides  of  a  quadrilateral   are   30   rds.,  40  rds., 
50  rds.,  and  60  rds.,  and  the  diagonal  drawn  from  the 
intersection  of  the  sides,  whose  lengths  are  30  rds.  and 
40  rds.,  is  70  rds. ;    required  the  area. 

Ans.  1874.22  sq.  rds. 

6.  The   sides  of  a  quadrilateral  are   25   rds.,  35  rds., 
45  rds.,  55   rds.,  and  the   angle   included   by  the  sides, 
whose  lengths  are  35  rds.  and  45  rds.,  is  50°;  required 
the  area.  A n s.  927.47  sq.  rds. 

7.  The  diagonals  of  a   quadrilateral   are  30  rds.  and 
40    rds.,    and   their    included    angle    is    30° ;     required 
the  area.  Ans.  300  sq.  rds. 

8.  The   angles  of  a   quadrilateral   are  80°,  110°,  88°, 
82°,  the  side  included  by  the  first  and  second  of  these 
angles   is  25   rds.,  and   the   side  included  by  the  third 
and  fourth  angles  is  45  rds. ;   required  the  area. 

Ans.  4105.08  sq.  rds.  ' 


SURFACES.  159 

9.  Three  sides  of  a  quadrilateral  are  20  rds.,  30  rds., 
40  rds.,  the  angle  included   by  the  first   and  second  is 
60°,  and  between  the  second  and  third,  80° ;    required 
the  area.  Ans.  593.58  sq.  rds. 

10.  The  sides  of  a  quadrilateral  inscribed  in  a  circle 
are  40  rds.,  50  rds.,  60  rds.,  70  rds. ;   required  the  area. 

Ans.  2898.28  sq.  rds. 

11.  The  area  of  a  parallelogram  is  47.055  sq.  ft.,  the 
sides  are  6  ft.  and  8  ft.;    required  the  diagonal. 

Ans.  9  ft.,  or  10.906  ft. 

12.  If   the    adjacent    sides  of  a   parallelogram   are   b 
and  c,  and  their  included  angle  A,  find  A  and  k  when 
&  is  a  maximum.  Ans.  A  =  90°,  k  =  be. 

13.  The  sides   and  angles  being  expressed  as  in  the 
last  example,  find  A  and  k  when  k  is  a  minimum. 

Ans.  A  =  0°  or  180°,  k  =  0. 

14.  If  only  two  adjacent   sides,  b  and   e,  of  a  paral- 
lelogram  be   given,  prove   that  k  is  indeterminate  be- 
tween the  limits  0  and  be. 

15.  Prove  that  the  diagonals  of  a  parallelogram  divide 
it  into  four  equal  triangles. 

164.    Problem. 

To  find  the  area  of  an  irregular  polygon. 

1.  When    the    sides    and    diagonals    from    the    same 
vertex  are  given. 

The  diagonals  divide  the   polygon 
into  triangles  whose  sides  are  given. 

The  areas  of  these  triangles,  &',  fc", 
&"',  . . .  are  found  by  article  160,  (5). 


160 


MENSURATION. 


2.  When  the  diagonals  from  the  same  vertex,  and 
the  perpendiculars  to  these  diagonals  from  the  oppo- 
site vertices  are  given. 


(2)     k  = 


3.  When  the  perpendiculars   to  a  diagonal  from  the 
vertices  of  the   opposite   angles   and  the   segments   of 
the  diagonal  made  by  these  perpendiculars  are  given. 

The  polygon     is    divided     into 

right  triangles     and     trapezoids, 

whose  areas   &',    fc",    &'",    ....    are 

found  by   article    162,    (2),    (4). 

(3)  &  =  V  +  Jb"  +  F'  +  . . . 

4.  When  one  side  of  a  figure  is  a  straight  line,  and 
the  opposite  side  is  an  irregular  curve  or  broken  line. 

vLet  the  straight  line  be  divided 
into  the  parts  «,  a',  a",  ....,   and      r\ 
let  the  perpendiculars  be  p,  </,  r,  . . . 
dividing    the   figure    into    parts    which    may    be    con- 
sidered trapezoids. 

(4)  k  =  4  a  (p  +  q)  +  i  «'(?  +  r)  +  }  a"(r  +  «). 
If  «'  ==  a  and  a" :  =  a,  (4)  becomes, 

(5)     fc  =  Ja(p-f27  +  2  r  -4-  s). 

165.    Examples. 

1.  Find  the  area  of  the  annexed 
polygon  if  p  =  10  rds.,  q  =  6  rds., 
r  =  6  rds.,  s  =  7  rds.,  t  =  15  rds., 
.d  =  14  rds.,  d'  =  16  rds.  Ana.  119.86  sq.  rds. 


SURFACES. 


161 


2.  Find   the  area  of  the   annexed 
polygon    if  p  =  3   rds.,    d  =  9  rds., 
p'  =  4    rds.,      dr   --       12    rds.,      and 
p"  ==  5  rds.  Am.  67.5  sq.  rds. 

3.  Find   the   area  of  the  annexed 
polygon  if  p  =  3  ft.,  p'  —  5  ft.,  p"= 
4  ft.,  a  =  5  ft.,   6  ==  6  ft.,  c  =  6  ft., 
d  =  9  ft.,   e=8ft.     Ans.  80.5  sq.  ft. 

4.  Find   the   area  of  the  annexed 
figure,     p  =  2  rds.,    q  =  3  rds.,   r  — 
4  rds.,  s  =  3  rds.,  a  =  a'  =  a"=  5  rds. 


47.5  sq.  rds. 
166.   Problem. 

To  find  the  area  of  a  regular  polygon. 

1.  When  the  perimeter  and  apothegm  are  given. 

Let  p  be   the   perimeter,   a   the   apo- 
them,    and   s  one  side  of  the  polygon. 

k  =  J  as  +  J  as  -f  J  as  -j-  J  as  +  ... 


.-.     (1)     fc  =  i 

2.  When   the   value   of   each    side    and    the    number 
of  sides  are  given. 

Let  s  be  one  side,  n  the  number 
of  sides,  a  the  apothem,  and  p  the 
perimeter. 

360°        180° 


p  =  ns.     DOB  = 
" 


-?- 

2  n 


OD  =  DB  cot  DOB,  or  a  = 

S.  N.  14. 


cot 


180C 


162 


MENSURATION, 


.-.     (2)     k  =  J ns2  cot 
If  s  ==  1,  then  (3)  k  =  J  w  cot 


180 


180< 


From  (3)  calculate  the  areas  of  the  regular  poly- 
gons each  of  whose  sides  is  1,  as  given  in  the  table 
subjoined 

167.  Table. 


Triangle  =  0.4330127. 
Square  =  1.0000000. 
Pentagon  ==  1.7204774. 
Hexagon  =  2.5980762. 
Heptagon  =  3.6339124. 


Octagon  ==  4.8284271. 
Enneagon  =  6.1818242. 
Decagon  7.6942088. 

Hendecagon^  9.3656399. 
Dodecagon  %  11.1961524. 


168.  Application  of  the  Table. 

Denoting  the  area  of  a  regular  polygon  whose  side 
is  s  by  A*,  and  the  area  of  a  similar  polygon  whose 
side  is  1,  as  given  in  the  table  by  k',  and  apply- 
ing the  principle  that  the  areas  of  similar  polygons 
are  to  each  other  as  the  squares  of  the  homologous 
sides,  we  have  the  proportion, 


k  :  k' 


I2. 


k  =  k's2. 


169.    Examples. 

1.  What   is  the   area   of  a   regular  hexagon  each  of 
whose  sides  is  6?  Ans.  93.5307432. 

2.  What  is  the  area  of  a  regular   pentagon  each  of 
whose  sides  is  10?  Ans.  172.04774. 


SURFACES. 


163 


3.  What   is   the   area   of   a   regular  decagon  each  of 
whose   sides   is   20?  Am.  3077.68352. 

4.  What  is  the  area  of  a  regular  dodecagon  each  of 
whose  sides   is    100?  An*.  111961.524. 

5.  What   is  the   area  of  a  regular  enneagon  each  of 
whose  sides  is  30?  Am.  5563.64178. 


170.    Formulas  for  the  Circle. 

Let  r  be   the   radius,  d   the   diameter,  c  the   circum- 
ference, and  k  the  area  of  a  circle,  then,  by  Geometry, 

we, have 

d  =--  2  r,     c  —  ird,     k  =  \rc. 

From  which  verify  the  following  table  of  formulas: 


2-  y-jfc 


3.  r 

4.  d  = 


-JI 

~vr 


5.     d  =  -~ 

7T 


7.  c  =  2  :rr. 

8.  c  =  Tfd. 

9.  c  =  2  l/Jbr! 

10.  Jk  =  Trr2. 

11.  fc  — 


12.     k  =  ^—- 


171.   Examples. 

1.  Given  the  radius  of  a  circle  —  10  rds. ;    required 
d,  c,  and  k. 

2.  Given  the  diameter  of  a  circle  —  20  rds. ;   required 
r,  c,  and  k. 


1  64  MEffiS  URA  TION. 

3.  Given   the   circumference   of  a  circle  =150  rds.  ; 
required  r,  d,  and  k. 

4.  Given   the    area   of   a   circle  —  1000  sq.  rds.;    re- 
quired r,  d,  and  c. 

5.  Find  the  diameter  of  a  circle  whose  area  is  equal 
to   that   of  a   regular   decagon,  each   side   of   which    is 
10  ft.  Ans.  31.3. 

6.  The  radius   of   a   circle  is  10  ft.,  the  diagonals  of 
an  equal  parallelogram  are  24  ft.  and  30  ft.;    required 
their  included  angle.  Ans.  60°  46'  17". 

7.  The   radii   of   two   concentric   circles  are  r  and  r'; 
find   the   area  of   the   ring    included   by  their  circum- 
ferences. Ans.  TT  (r  +  r')  (r  —  r). 

172.   Problem. 

To  find  the  area  of  a  sector  of  a  circle. 

Let  a  be  the  arc  of  a  sector,  d  the  de- 
grees in  the  arc,  r  the  radius,  and  k  the 
area. 

By  Geometry,         (1)     k  =  \  ra. 
-rrr  =  the  semi-circumference, 

=  the  arc  of  1°.      .  -  .  =  the  arc  of  d°. 


, 

k= 


173.    Examples. 

1.  Find  the   area  of  a   sector  whose   arc   is  40°  and 
radius  is  10  ft.  Ans.  34.907  sq.  ft. 

2.  Find  the  area  of  a  sector  whose  arc  is  60°  24'  30" 
and  radius  is  100  rds.  Ans.  5271  64  sq.  rds. 


'SURFACES...'  165 

3.  The   area  of  a  sector   is  345  sq.  ft.,  the   radius  is 
20  ft.;    required  the  arc.  Am.  98°  50'  06". 

4.  The  area  of  a  sector  is  1000  sq.  rds.,  the  arc  is  30° 
45';    required  the  radius.  Ans.  61.04  rds. 


\U.   Problem. 

To  find  the  area  of  a  segment  of  a  circle. 

Let  d  be  the  degrees  in  the  arc  of 
the  segment,  r  the  radius,  and  k  the 
area. 

By  the  last  problem,   .,.. 

—  the  area  of  the  sector. 
obU 

^r2  sin  d  =  the  area  of  the  triangle. 
dfrr2 

•'• 


If  d  is  greater  than  180,  sin  d  is  negative,  and  the 
second  term  in  the  value  of  k  becomes  positive,  as  it 
should,  since,  in  this  case,,  the  segment  is  equal  to 
the  corresponding  sector  plus  the  triangle. 


175.   Examples. 

1.  Find  the   area  of  the   segment  of  a  circle  whose 
arc  is  36°  and  radius  10  ft.  Ans.  2.027  sq.  ft. 

2.  Find  the  area  of  a  segment  whose  chord  is  36  ft. 
and  radius  30  ft.  Ans.  147.30  sq.  ft. 

3.  Find  the  area  of' a  segment  whose  altitude w  is  36 
rds.  and  radius  50  rds.  Ans.  2545.85  sq.  rds. 


166 

4.  The  area  of  a  segment  is  2545.85  sq.  rds.,  the 
radius  is  50  rds.;  required  the  number  of  degrees  in 
the  arc.  .  . 

176.    Problem. 

To  find  the  area  of  an  ellipse. 

Let  a  be  the  semi-major  axis,  and  b  the 
semi-minor  axis! 

Then,   Ray's    Analytic   Geometry,  article   446, 

k  =  TTob. 


177.    Examples. 

1.  The  semi-axes  of  an  ellipse  are  10  in.  and  7  in. ; 
required  the  area.  An#.  219.912  sq.  in. 

2.  The   area  of  an   ellipse   is   125   sq.  rds.;    find  the 
axes  if  they  are  to  each  other  as  3  is  to  2. 

Am.  15.45;  10.30. 


178.    Problem. 

To  find   the  area  of  the   entire   surface  of  a   right  prism 

Let  p  be  the  perimeter  of  the  base, 
a  the  altitude,  s  one  side  of  the  base, 
k'  the  area  of  a  polygon  similar  to  the 
base,  each  side  of  which  is  unity,  ar- 
ticle 167,  and  k  the  area  of  the  entire 
.surface. 

ap  =  the  convex  surface. 
2  k's*  =  the  areas  of  the  bases.     Article  168. 
.  • .     k  =  ap  -f  2 


'SURFACES:  167 

179.  Examples. 

1.  What  is  the  entire  surface  of  a  right  prism  whose 
altitude  is  20  ft.,  and  base  a  regular  octagon  each  side 
of  which  is  10  ft.?  Ans.  2565.68542  sq.  ft. 

2.  What    is   the   entire  surface  of   a  right  hexagonal 
prism  whose   altitude    is    12   ft.,  and   each   side   of   the 
base  is  6  ft.?  Ana.  619.0614864  sq.  ft. 

3.  What  is  the  entire  surface  of  a  right  prism  whose 
altitude  is  15  in.,  and  base  a  regular  triangle  each  side 
of  which  is  3  in.?  Am.  142.7942286  sq.  in. 

180.  Problem. 

To  find  the  area  of  the  surface  of  a  regular  pyramid. 

Let  p  be  the  perimeter  of  the  base,  a 
the  slant  height,  s  one  side  of  the  base, 
k'  and  k  as  in  the  last  problem. 

±ap  —  the  convex  surface. 
k's2  —   the  area  of  the  base. 


181.    Examples. 

1.  What  is  the  entire   surface   of  a  regular  pyramid 
whose   slant   height   is   12   ft.,  and   base   a   regular  tri- 
angle  each   side   of   which   is  5   ft.  ? 

Ans.  100.82532  sq.  ft. 

2.  What    is  the  entire    surface  of  a  right   pyramid 
whose  slant  height  is  100  ft.,  and  base  a  regular  deca- 
gon each -side  of  which  is  20  ft.? 

Am.  13077.68352  sq.  ft. 


168 


MENSURATION. 


182.    Problem. 

To  find  the  entire  surface  of  a  frustum  of  a  right  pyramid. 

Let  p  be  the  perimeter  of  the  lower 
base,  p'  the  perimeter  of  the  upper 
base,  a  the  slant  height,  s  one  side  of 
the  lower  base,  s'  one  side  of  the  upper 
base,  kf  and  k  as  in  Art.  178. 

$a(p  -f  p')  —  the  convex  surface. 

k's2  =  the  area  of  lower  base. 
k's'2=  the  area  of  upper  base. 


183.   Examples. 

1.  What    is    the    entire    surface   of   a   frustum   of   a 
pyramid  whose   slant  height   is   12   ft.,   an-d   the   bases 
regular   decagons  whose   sides   are   8   ft.  and   5    ft.,   re- 
spectively? Ans.  1464.78458  sq.  ft. 

2.  What    is    the    entire   surface    of    a   frustum    of    a 
pyramid   whose   slant   height   is   15   ft.,  and   the   bases 
regular   hexagons  whose   sides   are   10  ft.  and  6  ft.,  re- 
spectively? Ans.  1073.338  sq.  ft. 

184.   Problem. 

To  find  the  area  of  the  entire  surface  of  a  cylinder. 

Let  r  be  the  radius  of  the  cylinder, 
a  its  altitude,  and  k  the  area  of  the 
entire  surface. 

2  nra  =  the  convex  surface. 
2  -rrr2  =  the  area  of  the  bases. 

.-.     k  =  2  TIT  (a  +  r). 


SURFACES.  169 

185.    Examples. 

1.  What   is  the   entire    surface  of  a  cylinder  whose 
altitude   is   6  ft.  and   radius   2   ft.? 

Am.  100.5312  sq.  ft. 

2.  What   is   the   entire    surface   of   a   cylinder   whose 
altitude  is  100  ft.  and  radius  20  ft.? 

Am.  15079.68  sq.  ft. 

186.    Problem. 

To  find  the  area  of  the  entire  surface  of  a  rone. 

Let  r  be  the  radius  of  the  base  of  the 
cone,  a  the  slant  height,  and  k  the  area 
of  the  entire  surface. 

-rrra  =  the  convex  surface. 
Trr2  =  the  area  of  the  base. 

.  • .     k  =  TIT  (a  -f  r). 

187.    Examples. 

1.  What  is  the  entire  surface  of  a  cone  whose   slant 
height  is  10  ft.  and  radius  5  ft.?     Am.  235.62  sq.  ft. 

2.  What  is  the  entire  surface  of  a  cone  whose   alti- 
tude is  100  ft.  and  radius  25  ft.  ? 

Ans.  10059.1675  sq.  ft. 

188.    Problem. 

To  find   the   area   of  the  entire  surface   of  the  frustum 
of  a  cone. 

Let    r   be    the    radius   of   the    lower   base,  /  be   the 

S.  N.  15. 


170  MENSURATION. 

radius  of  the   upper   base,  a   the    slant   height,  and   k 
the   area  of   the  entire  surface. 

na  (r  -j-  r' )  —  the  convex  surface. 

77T2  =  the  area  of  the  lower  base. 
7T/2  =  the  area  of  the  upper  base. 

.;.     k  ==  7r[a(r-hr')-f-r2 

189.    Examples. 

1.  Find  the  entire  surface  of   the  frustum  of   a  cone 
of  which   the   radius  of   the   lower   base   is   10  ft.,  the 
radius   of   the   upper   base    is   6   ft.,  and   slant   height 
is  20  ft.  An*.  1432.5696  sq.  ft. 

2.  Find  the  entire  surface  of  the  frustum  of  a  cone 
of  which   the   radius  of  the   lower   base  is  25  in.,  the 
radius  of  the  upper  base   12   in.,  and  the  slant  height 
36  in.  Ans.  45.8368  sq.  ft. 

190.   Problem. 

To  find  the  area  of  the  surface  of  a  sphere. 

Let  r  be  the   radius,  d  the  diameter,  c  the   circum- 
ference, and  k  the  area.         Then,  by  Geometry, 
(1)     k  =  4nr2.       (2)     k  =  nd2. 

(3)     k  =  ^--          (4)     k  =  cd. 

191.   Examples. 

1.  The   radius   of   a   sphere   is   10  ft.;    required   the 
area.  Ans.  1256.64  sq.  ft. 

2.  The  diameter  of  a  sphere  is  25  ft. ;   required  the 
area.  Ans.  1963.5  sq.  ft. 


SURFACES.  171 

3.  The  circumference  of  a  sphere  is  100  in.;   required 
the  area.  Ans.  3183.0914  sq.  in. 

4.  The  circumference  of  a  sphere   is    62.832,  and  di- 
ameter 20;   required  the  area.  Ans.  1256.64. 

192.    Problem. 

To  find  the  area  of  a  zone. 

By  Geometry,  the  area  of  a  zone  is  eq'ual  to  the  cir- 
cumference of  a  great  circle  multiplied  by  the  altitude 
of  the  zone. 

Let  a  denote  the  altitude  of  the  zone,  r  the  radius 
of  the  sphere,  and  k  the  area  of  the  zone. 

.  • .     k  =  2  Trra, 

193.    Examples. 

1.  What   is  the   area  of  the  torrid  zone,  calling  its 
width  46°  56',  and  the  earth   a  perfect  sphere  whose 
radius  is  3956.5  mi.?  Ans.  78333333.  sq.  mi. 

2.  What  is  the  area  of  the   tv/o  frigid  zones  if  the 
polar  circles  are  23°  28'  from  the  poles? 

Ans.  16270370.  sq.  mi. 

3.  What  is  the  area  of  the  two  temperate  zones? 

Ans.  102109933.  sq.  mi. 

194.   Problem. 

To  find  the  area  of  a  spherical  triangle. 

Let   s  =  A  +  B  -f- <?,   and  ^Trr2  =  the 
tri-rectangular  triangle. 

Then,  by  Geometry, 

I  774*2    (  _£ O  \ 

90°       Z)' 


1 72  MENS  URA  TIOX. 

In  this  formula,  7^  --2   is   to   be    regarded    as   an 
yu 

abstract   number.     Minutes   and   seconds  are   to   be  re- 
duced to  the  decimal  of  a  degree. 

195.    Examples. 

1.  Find   the    area    of    the    spherical    triangle   whose 
angles  are  60°,  80°,  100°,  and  the  radius  3956.5  mi. 

Am.  16392592  sq.  mi. 

2.  Find  the  area  of  a  spherical  triangle  whose  sides 
are  70°,  90°,  100°,  respectively,  and  radius  100  in. 

Ans.  10942.1928  sq.  in. 

196.   Problem. 

To  find  the  area  of  a  spherical  polygon. 

Let  s  be  the  sum  of  the  angles,  n  the  number  of 
sides,  k  the  area  of  the  polygon,  and  r  the  radius  of 
the  sphere. 

Then,  by  Geometry, 


197.    Examples. 

1.  The  sum  of  the  angles  of  a  spherical  hexagon  is 
800°,  the  radius  is  100  ft. ;   required  the  area. 

Ans.  13963.  sq.  ft. 

2.  Each  angle  of  a  spherical  pentago.n  is  120°,  the 
radius  is  50  ft. ;  required  the  area.       -4ns.  2618.  sq.  ft. 


SURFACES.  173 

3.  The  angles  of  a  spherical   polygon  are  90°,  100°, 
110°,  150°,  respectively,  the   radius  is  10  ft.;    required 
the  area.  Ans.  157.08  sq.  ft. 

4.  Each   angle   of  a  spherical  decagon   is    150°,  the 
radius  is  1  ft.;   required  the  area.          Ans.  1.0472  ft. 


198.   Problem. 

To  find  the  area  of  the  surface  of  a  regular  polyhedron. 

Let  e  be  one  edge,  n  the  number  of  faces,  k'  the 
area  of  a  polygon  whose  side  is  1,  and  similar  to  one 
face,  and  k  the  area  of  the  entire  surface. 

k'e2  —  the  area  of  one  face.     Article  168. 


199.    Examples. 

1.  What  is  the  area  of  the  entire  surface  of  a  tetra- 
hedron whose  edge  is  10  ft.?        Ans.  173.20508  sq.ft. 

2.  What  is  the  area  of  the  entire  surface  of  a  hexa- 
hedron whose  edge  is  5  ft.?  Ans.  150  sq.  ft. 

3.  What  is  the  area  of  the  entire  surface  of  an  octa- 
hedron whose  edge  is  20  ft.?      Ans.  1385.64064  sq.  ft. 

4.  What  is  the  area  of  the  entire  surface  of  a  dodec- 
ahedron   whose   edge  is  15  in.?      A  ns.  32.25895  sq.ft. 

5.  What  is  the  area  of  the  entire  surface  of  an  icosa- 
hedron  whose  edge  is  100  in.  ?      Ans.  601.4065  sq.  ft. 


174  MENSURATION. 

MENSURATION   OF  VOLUMES. 

200.  Problem. 

To  find  the  volume  of  a  prism. 

Let  k  be  the  area  of  the  base,  a  the  altitude,  and  v 
the  volume.     Then,  %  Geometry, 

v  =  ak. 

201.  Examples. 

1.  What  is  the  volume  of  a  regular  hexagonal  prism 
whose  altitude  is  20  ft.,  and  each  side  of  the  base  10  ft.? 

Am.  5196.1524  cu.  ft. 

2.  What  is  the  volume  of  a  triangular  prism  whose 
altitude   is   6  ft.,  and   the   sides  of  its  base  3  ft.,  4  ft., 
and  5  ft.,  respectively?  Ans.  36  cu.  ft. 

3.  What  is  the  volume  of  a  regular  octagonal  prism 
whose  altitude    is    120  ft.,  and  each   side  of  the  base 
20  ft.?  Ans.  231764.5008  cu.  ft. 

202.    Problem. 

To  find  the  volume  of  a  pyramid. 

Let  k  be  the  area  of  the  base,  a  the  altitude,  and  v 
the  volume. 


203.   Examples. 

• 

1.  What  is  the  volume  of  a  pyramid  whose  altitude 

is   15  ft.,  and  whose  base  is  a  regular  heptagon  each 
side  of  which  is  5  ft.?  Ans.  454.23905  cu.  ft. 

2.  What  is  the  volume  of  a  pyramid  whose  altitude 
is  21   in.,  and  whose  base  is  a  triangle   each    side  of 
which  is  30  in.?  Ans.  2727.98  cu.  in. 


VOLUMES.  175 

204.  Problem. 

To  find  the  volume  of  the  frustum  of  a  pyramid. 

Let  k  and  kl  be  the  areas  of  the  bases,  a  the  alti- 
tude, and  v  the  volume.  Then,  by  Geometry, 

(1)  v  =  $a(k  +  kl+V~kk;'). 

If  the  bases  are  regular  polygons  whose  .sides  are 
s  and  s',  we  shall  have,  by  article  168,  k  =  &'s2,  and 
fcj  =  k's'2,  in  which  k'  is  given  in  the  table  of  article 
167,  and  (1)'  becomes 

(2)  v  =  J-  a  (s2  -f  s'2  +  ss')  kf. 

205.  Examples. 

1.  What  is  the  volume  of  the  frustum  of  a  pyramid 
whose   altitude   is   9   ft.,  and  whose    bases   are   regular 
triangles,  one   side   of  the   lower   being   8   ft.,  and  one 
side  of  upper,  5  ft.?  Ann.  167.576  cu.  ft. 

2.  What  is  the  volume  of  the  frustum  of  a  pyramid 
whose   altitude   is   27   in.,  and  the  bases  regular  hexa- 
gons, the  sides  of  which  are  10  in.  and  6  in.,  respect- 
ively? '  An*.  4583.0064  cu.  in. 

206.  Problem. 

To  find  the  volume  of  a  'cylinder. 

Let  r  represent  the  radius,  a  the  altitude,  and  v  the 
volume- 


207.    Examples. 

1.  What  is  the  volume  of  a  cylinder  whose  altitude 
is  50  in.,  and  radius  15  in.?  Ans.  20.453  cu.  ft. 

2.  What  is  the  volume  of  a  cylinder  whose  altitude 
is  25  ft.,  and  radius  4  ft.?  Ans.  1256.64  cu.  ft. 


176  MENSURATION. 

208.    Problem. 

To  find  the  volume  of  a  cone. 

Let   r   be   the   radius  of   the   base,   a  the   altitude, 

and  v  the  volume. 

v  =     arrr2. 


.   Examples. 

1.  What  is  the  volume  of  a  cone  whose  altitude   is 
21  in.,  and  radius  10  in.?  Am.  2199.12  cu.  in. 

2.  What  is  the  volume  of  a  cone  whose  altitude   is 
30  ft.,  and  radius  is  10  ft.?  Ans.  31416.  cu.  ft. 

210.   Problem. 

To  find  the  volume  of  the  fnmtum  of  a  rone. 

Let  r  and  r  be  the  radii  of  the  bases,  n  the  altitude, 
and  v  the  volume. 

v  =  Jcwr(r2-|-r'2+rrf). 

f 

211.    Examples. 

1.  What  is  the  volume  of  the  frustum  of  H  cone 
whose  altitude  is  15  ft.,  and  the  radii  of  whose  bases 
are  9  ft.  and  4  ft,  respectively?  Am.  2089.164  cu.  ft. 

±  How  many  barrels  will  that  cistern  contain  wlmsc 
altitude  is  s  ft.,  the  diameter  at  the  bottom  4  ft.,  and 
at  the  top  (i  ft.?  Am.  37.8  bbl. 

212.    Formulas  for  the  Sphere. 

Let  r  be  the  radiu>.  <l  the  diameter,  <•  the  oircum- 
ference,  k  the  area  of  the  surface,  and  r  the  volume 


VOLUMES.  177 

of  a  sphere,  then,  by  Geometry,  we  have 

d  —  2  r,     £  =  nd,     k  =  4  Try2,     v  =  ^rk. 
From  which  verify   the    following    table   of   formulas: 


1.  r  ==  $d. 

2.  r  =  J5_. 

3.  r  = 


4.     r  =  \  ;  _2 


5.     (2  ±:=  2  r. 


7.  fc/?,^ 

8.  d  =  III1 


9.     r  ^  2  rrr. 
10.     r-         rr(/. 


11.  c  ^  V-nk. 

13.  k  =  4  7rr2. 

14.  k  =  rrrf2. 

15.  jfc^-2-. 

7T 

16.  Jb  =  ^367ri 

17.  0  =  t7ryS. 

18.  |7r=r|-7Trf3. 

19.  *  =  g^. 

20.  t;  =="Jti/Z 


Examples. 

1.  Calling   the   diameter  of  the  earth   7913  mi.,  and 
the  diameter  of  the  sun  856,000,  find  the  ratio  of  their 
surfaces,  also  the  ratio  of  their  volumes. 

2.  What    is    the    volume   of    the   shell   of    a    hollow 
>*phere  whose   radius   is  8  ft.  4   in.,  and  the  thickness 
of  the  shell  3  ft.  6  in.?  Am.  1951.1081  cu.  ft. 


178 


MENSURATION. 


214.    Problem. 

To  find  the  volume  of  a  spherical  sector. 

A  spherical  sector  is  the  volume  generated  by  the 
revolution  of  any  circular  sector,  ABC, 
about  any  diameter,  DE.  By  Geometry, 
the  volume  of  a  spherical  sector  is 
equal  to  the  zone  which  forms  its  base, 
multiplied  by  one-third  of  the  radius. 

Let  a  be  the  altitude  of  the   zone, 
and  r  the  radius. 

.  • .    v  =  §  7rr2a. 


215.    Examples. 

1.  The   altitude   of  the    zone  which  forms  the  base 
of  a  sector   is   6  ft.,  the  radius  is  12  ft.;    required  the 
volume.  Ans.  1809.5616  cu.  ft. 

2.  The  angle  BCD,  in  the  diagram  of  last  article,  is 
20°,  ACS  is  35°,  r  =  20  ft. ;    required  the  volume. 

Ans.  6134.25  cu.  ft. 


216.   Problem. 

'To  find  the  volume  of  a  spherical  segment. 

A  spherical   segment  is  the  portion  of  a  sphere  in- 
cluded between  two  parallel  planes. 

Let   r'  -  -  BF  perpendicular  to  DE, 
and  r"  —  AG  perpendicular  to  DE. 

r  =  the  radius,  d'=  CF,  and  d"  =  CG. 
v  —  the  vol.  generated  by  ABFG. 
tf=  the  vol.  generated  by  ABC=%Trr2a. 


VOLUMES.  179 


v"=  the  vol.  generated  by  BFC  = 

t/"=  the  vol.  generated  by  ,4GC  =  JrfW2. 

v  =  v'+v"^vm. 

The   sign   of  vm  is  —  or  -f  according   as  AG    is   on 
the  same  or  opposite  side  of  the  center  as  BF. 
.  •  .     v  --=  J-  TT  (2  ar2  -f-  dVa  +  d'Y"2). 

217.  Examples. 

1.  r  =  12  in.,   r'—  3  in.,   r"  =  10  in.;    required  v. 

2.  Two  parallel  planes  divide  a  sphere  whose  diame- 
ter is  36   in.  into  three  equal  segments;    required   the 
altitude  of  each.  An*.  13.93  in.;  8.14  in.;  13.93  in. 

218.  Problem. 

To  find   the  volume  generated  by  the  revolution  of  a  cir- 
cular segment  about  a  diameter  exterior  to  it. 

Let  'v  =  vol.  generated  by  ADB. 
v'  =  vol.  generated  by  ADBC. 
v"=  vol.  generated  by  ABC. 

'v  =  v'  —  v". 
Let  a  —  FGj  c  =  AB,  p  —  C7,  perpendicular  to  AB. 

v'  =  §  Trar2,         v"  =  f  nap2. 
.-.     v'—v"=%  TTO  (r2  —  jo2)  =  J  Trac3. 


219.   Examples. 

1.  a  —  -  5  in.,  c  =  8  in.;  find  v.     ^16-.  167.552  cu.  In. 

2.  A  sphere  6  in.  in  diameter   is   bored   through  the 
center  with   a  3-inch   auger  ;    required   the  volume   re- 
gaining. Ans.  73.457  cu.  in. 


180  MENSURATION. 

3.  Prove  that  the  volume  generated  by  the  segment 
whose  altitude  is  a  and  chord  c  is  to  the  sphere  whose 
diameter  is  c  as  a  :  c. 

4.  Prove   that  if  c  is  parallel  to  the  diameter  about 
which    it    is    revolved,   the   volume    generated   by   the 
segment    is   equal    to   the  volume   of   a   sphere   whose 
diameter  is  c. 


2-20.    Problem. 

To  find  the  volume  of  a  ivedge. 

The  base  is  a  rectangle,  the  sides  are  trapezoids,  the 
ends,  triangles. 

Let  e  be  the  edge,  I  the  length 
of  base,  b  the  breadth  of  base, 
and  a  the  altitude.  VXl""  l  X' 

Passing    planes    through    the 

extremities  of  the  edge  perpendicular  to  the  base,  we 
have  a  triangular  prism  and  two  pyramids.  These 
pyramids  may  fall  within  or  without  the  wedge,  or 
one  or  both  of  the  pyramids  may  vanish. 

But  in  all  cases  the  formula  is  the  same. 


=  the  volume  of  the  prism. 
a  (I  —  e)  b  =  the  volume  of  the  pyramids. 


221.   Examples. 

1.  The  edge  of  a  wedge  is  6  in.,  the  altitude  12  in., 
the  length  of  base  9  in.,  and  the  breadth  of  base  5  in.; 
what  is  the  volume  ?  Ans.  240  cu.  in. 


VOLUMES.  181 

.  The  edge  of  a  wedge  is  20  ft.,  the  altitude  24  ft., 
length  of  base   15  ft.,  the  breadth  of  base  10  ft.  ; 
what  is  the  volume  ?  Ans.  2000  cu.  ft. 

22'2.    Problem. 

To  find  the  volume  of  a  rectangular  prismoid. 

The  bases  are  parallel  rect- 
angles, the  other  faces  are 
trapezoids. 

Let  /  and  b  be  the  length 
and  breadth  of  the  lower  base, 
/'  and  6'  the  length  and  breadth 
of  the  upper  base,  and  a  the 
altitude. 

Passing    the    plane    as  .  indicated,    the    prismoid    is 
divided  into  t\vo  wedges. 

J  06  (2  I  +  I')  ~  the  vol.  of  wedge  whose  base  is  bl. 
b'(2  I'  +  0  ==  the  vol.  of  wedge  whose  base  is  bT. 


223.   Examples. 

1.  The   length   and   breadth  of  the   lower   base  of  a 
rectangular  prismoid  are  25   ft.  and   20  ft.,  the   length 
and   breadth   of   the   upper  base   are    15   ft.  and  10  ft., 
and  the  altitude  is  18  ft.;    what  is  the  volume? 

Ans.  5550  cu.  ft. 

2.  The   length   and   breadth  of  the   lower   base  of  a 
rectangular    prismoid    are    15    yds.    and    10    yds.,    the 
length    and    breadth    of    the    upper    base    are    9    yds. 
and   6  yds,,  and  the   altitude  is  18   yds.;    what  is  the 
volume?  Ans.  1764  cu.  yds. 


182  MENSURA  TIGS. 


224.    Problem. 

To  find  the  dihedral  angle  included  by  the  faces  of  a 
regular  polyhedron. 

Conceive  a  sphere  whose  radius  is  1  so  placed  that 
its  center  shall  be  at  any  vertex  of  the  polyhedron. 

The  faces  of  the  polyhedral  angle  will  intersect  the 
surface  of  the  sphere  in  a  regular  polygon^  whose  sides 
measure  the  plane  angles  that  include  the  polyhedral 
angle,  and  whose  angles  are  each  equal  to  the  required 
dihedral  angle. 

Let  ABCD  be  such  a  polygon,  P 
the  pole  of  a  small  circle  passing 
through  A,  B,  C,  D,  E.  Join  P  with 
the  vertices  and  with  the  middle  of 
AB  by  arcs  of  great  circles. 

Let  n  denote  the  number  of  sides  of  the  polygon, 
s  =  one  side,  and  A  =  a  dihedral  angle. 

360°        180° 

.  • .     APQ  =  - —  ==£—.,  and  A  Q  —  ^  s. 
2  n  n 

By  Napier's  circular  parts,  we  have 

sin  (90°  —  APQ)  =  cos  AQ  cos  (90°  —  PAQ). 

1SO° 
or  sin  (90°—       -)  =  cos  £*   cos  (90°—  $  A). 

n    ' 

180° 

or  cos  -          =  cos  IM  sin  \A. 
n 

cos  1180° 
. '.     sin  \  A  =  - 

cos  -J-s 

In  the  Tetrahedron,  n  ==  3,     and    s  =  60°, 

*  i  COS     Ov/  »~/~\n    n-4/     inn 

•'•    Sln       =  :-     •'•    A='°  31  42' 


VOLUMES.  183 

In  the  Hexahedron,    n  =  3,  and   s  =  90°, 
•     i  A  —  cos  6Q°  .        j  _  ono 

In  the  Octahedron,      n  =  4,  and    s  =  60°, 

.  • .     sin  J  yl  =  C  M~  •          • '  •     ^  =  109°  28'  18". 

In  the  Dodecahedron,  n  =  3,  and    s  =  108°, 

.-.    sin  \A=  C°S  ??°.          .-.     ^  =  116°  33r  54". 
cos  54° 

In  the  Icosahedron,    n  =  5,  and    s  =  60°, 

.-,    sin  \ A  =  COS  ^!-          .-.     4=  138°  11'  23/r. 


225.   Problem. 

To  find  the  volume  of  a  regular  polyhedron. 

If  planes  be  passed  through  the  edges  of  the  poly- 
hedron and  the  center,  they  will  bisect  the  dihedral 
angles  and  divide  the  polyhedron  into  as  many  pyra- 
mids as  it  has  faces.  The  faces  will  be  the  bases  of  the 
pyramids,  the  center  will  be  their  common  vertex,  the 
line  drawn  from  the  center  of  the  polyhedron  to  the 
center  of  any  base  will  be  perpendicular  to  the  base, 
and  will  be  the  altitude  of  the  pyramid. 

From  the  foot  of  the  perpendicular  draw  a  perpen- 
dicular to  one  side  of  the  base,  and  join  the  foot  of 
this  perpendicular  with  the  center.  We  thus  have  a 
right  triangle  whose  perpendicular  is  the  altitude  of 
the  pyramid,  the  base  the  apothem  of  one  face  of  the 
polyhedron,  the  angle  opposite  the  perpendicular  one- 
half  the  dihedral  angle  of  the  polyhedron. 


184 


MENSURATION. 


Let  p  be  the  perpendicular,  a  the  apothem  of  one 
face,  J  A  one-half  of  a  dihedral  angle,  n'  the  number 
of  sides  of  one  face,  and  e  one  edge. 

p  =  a  tan  $  A,     a  =  \e  cot  -7- 180°.     Article  166. 

.  • .    p  =  \e  cot  ^180°  tan  \  A. 
Let  &',  w,  and  k  be  the  same  as  in  article  198. 
Then,  ^pk  =  the   volume  of  the  polyhedron. 

.».    v  =  J^fc'e3  cot  ^180°  tan  \A. 
Let  c  —  1,  and  verify  the  table  subjoined  : 

226.   Table. 


Names. 

Surfaces. 

Volume. 

Tetrahedron 

1.7320508 

0.1178513 

Hexahedron 

6.0000000 

1.0000000 

Octahedron 

3.4641016 

0.4714045 

Dodecahedron 

20.6457288 

7.6631189 

Icosahedron 

8.6602540 

2.1816950 

227.    Application  of  the  Table. 

Let    i''    and    v    denote    similar    regular    polyhedrons 

whose  edges  are  1  and  e,  respectively.    Then  we  have 

r'  :  c  :  :  I3   :  e3.         .;. 


228.    Examples. 

1.   What  is  the  volume  of  a  tetrahedron  whose  edge 
is  10  ft.?  -I/,*.  117.8513  cu.  ft. 


2.  The  volume  of  a  hexahedron  is  134217728  cu.  in. 


what  is  its  surface? 


Ana.  1572864  sq.  in. 


SURVEYING. 

229.    Definition  and  Classification. 

Surveying  is  the  art  of  laying  out,  measuring,  and 
dividing  land,  and  of  representing  on  paper  its  bound- 
aries and  peculiarities  of  surface. 

There  are  three  branches — Plane,  Geodesic,  and  Topo- 
graphical. 

Plane  surveying  is  that  branch  in  which  the  por- 
tion surveyed  is  regarded  as  a  plane,  as  is  the  case  in 
small  surveys. 

Geodesic  surveying  is  that  branch  in  which  the  curva- 
ture of  the  surface  of  the  earth  is  taken  into  consider- 
ation, as  is  the  case  in  all  extensive  surveying. 

Topographical  surveying  is  that  branch  in  which  the 
slope  and  irregularities  of  the  surface,  the  course  of 
streams,  the  position  and  form  of  lakes  and  ponds,  the 
situation  of  trees,  marshes,  rocks,  buildings,  etc.,  are 
considered  and  delineated. 

INSTRUMENTS. 

230.    Classification. 

The  instruments  employed  in  surveying  may  be 
classed  as  Field  instruments  and  Plotting  instruments. 

The  principal  field  instruments  are  the  chain  and 
tally  pins,  marking  tools,  field-book  and  pencil,  the  magnetic 

S.  N.  16.  (185) 


186  SURVEYING. 

compass,  the  solar  compass,  the   transit  compass,  the  level, 
and  the  theodolite. 

The  principal  plotting  instruments  are  the  dividers, 
the  ruler  and  triangle,  parallel  rulers,  the  diagonal  scale, 
the  semicircular  protractor. 

231.   The  Chain  and  Tally  Pins. 

The  chain  is  4  rods  or  66  feet  in  length,  and  is  di- 
vided into  100  links,  each  equal  to  7.92  inches. 

After  every  tenth  link  from  each  end  is  a  piece  of 
brass,  notched  so  as  to  indicate  the  number  of  links 
from  the  end  of  the  chain,  thus  facilitating  the  count- 
ing of  the  links. 

A  half-chain  of  50  links  is  sometimes  used,  especially 
in  rough  or  hilly  districts. 

The  tally  pins  are  made  of  iron  or  steel,  about  12 
inches  in  length  and  one-eighth  of  an  inch  in  thick- 
ness, heavier  toward  the  point,  with  a  ring  at  the  top 
in  which  is  fastened  a  piece  of  cloth  of  some  con- 
spicuous color. 

These  pins  are  conveniently  carried  by  stringing 
them  on  an  iron  ring  attached  to  a  belt  which  is 
passed  over  the  right  shoulder,  leaving  the  pins  sus- 
pended at  the  left  side. 

In  Government  surveys  eleven  tally  pins  are  used. 

232.   Marking  Tools. 

A  surveying  party  will  need  an  ax  for  cutting 
notches,  cutting  and  driving  stakes  and  posts;  a  spade 
or  mattock  for  planting  or  finding  corners;  knives,  or 
other  tools,  for  cutting  letters  or  figures;  and  a  file 
and  whetstone  for  keeping  the  tools  in  order. 


INSTRUMENTS.  187 

233.  Field-Book  and  Pencil. 

In  ordinary  practice  one  field-book  will  be  sufficient; 
but  in  surveying  the  public  lands,  four  different  books 
are  required  —  one  for  meridian  and  base  lines,  another 
for  standard  parallels  or  correction  lines,  another  for 
exterior  or  township  lines,  and  another  for  subdivision 
or  section  lines,  as  designated  on  the  title-page. 

A  good  pencil,  number  2  or  3,  well  sharpened,  should 
be  used,  so  that  the  notes  may  be  legible. 

A  temporary  book  may  be  used  on  the  ground,  and 
the  notes  taken  with  a  pencil.  These  notes  can  then 
be  carefully  transcribed  with  pen  and  ink  into  the 
permanent  field-book. 

234.  The  Magnetic  Compass. 

The  vernier  magnetic  compass  is  exhibited  in  the 
drawing  on  page  189. 

The  needle  turns  freely  on  a  pivot  at  the  center,  and 
settles  in  the  magnetic  meridian. 

The  compass  circle  is  divided,  on  its  upper  surface, 
to  half-degrees,  numbered  from  0°  to  90°  each  side  of 
the  line  of  zeros. 

The  sight  standards  are  firmly  fastened  at  right 
angles  to  the  plate  by  screws,  and  have  slits  cut 
through  nearly  their  whole  length,  terminated  at  in- 
tervals by  apertures  through  which  the  object  toward 
which  the  sights  are  directed  can  be  readily  found. 

Two  spirit  levels  at  right  angles  to  each  other  are 
attached  to  the  plate. 

Tangent  scales  are  scales  on  the  right  and  left  edges 
of  the  north  sight  standard,  the  one  on  the  right  be- 


188  SURVEYING. 

ing  used   in  taking   angles  of  elevation,  and  the  one 
on  the  left  in  taking  angles  of  depression. 

Eye-pieces  are  placed  on  the  right  and  left  sides  of 
the  south  sight  standard  —  the  one  on  the  right  near 
the  bottom,  the  one  on  the  left  near  the  top  —  each 
on  a  level,  when  the  compass  is  level,  with  the  zero 
of  its  tangent  scale.  These  eye-pieces  are  centers  of 
arcs  tangent  to  the  tangent  scales  at  the  zero  point. 

The  vernier  is  a  scale  movable  by  the  side  of  another 
scale,  and  divided  into  parts  each  a  little  greater  or 
a  little  less  than  a  part  of  the  other,  and  having  a 
known  ratio  to  it.  In  the  drawing  the  vernier  is 
represented  on  the  plate  near  the  south  sight. 

The  needle  lifter  is  a  concealed  spring,  moved  from 
beneath  the  main  plate,  by  which  the  needle  may  be 
lifted  to  avoid  blunting  the  point  of  the  pivot  in 
transporting  the  instrument. 

The  out-keeper  is  a  small  dial  plate,  having  an  index 
turned  by  a  milled  head,  and  is  used  in  keeping  tally 
in  chaining. 

The  ball  spindle  is  a  small  shaft,  slightly  conical,  to 
which  the  compass  is  fitted,  having  on  its  lower  end 
a  ball  confined  in  a  socket  by  a  light  pressure,  so  that 
the  ball  can  be  moved  in  any  direction  in  leveling 
the  instrument. 

The  clamp  screw  is  a  screw  in  the  side  of  the  hol- 
low cylinder  or  socket,  which  fits  to  the  ball  spindle, 
by  which  the  compass  may  be  clamped  to  the  spindle 
in  any  position. 

A  spring  catch,  fitted  to  the  socket,  slips  into  a 
groove  when  the  instrument  is  set  on  the  spindle, 
and  secures  it  from  slipping  from  the  spindle  when 
carried. 


190  SURVEYING. 

The  Jacob  staff  is  a  single  staff  to  support  the  com- 
pass, about  5J  feet  long,  having  at  the  upper  end 
the  ball  and  socket  joint,  and  terminating  at  the 
lower  end  in  a  sharp  steel  point,  so  as  to  be  set 
firmly  in  the  ground. 

The  tripod  is  a  three-legged  support  sometimes  used 
instead  of  the  Jacob  staff. 


235.    Adjustments  of  the  Compass. 

1.  To  adjust  the  level, —  Bring  the  bubbles  to  the  cen- 
ter of  the  tubes  by  pressing  the  plates  so  as  to  turn  the 
ball   slightly  in   its   sockets.     Turn    the   compass    half- 
way round,  and   if  either   bubble   runs   to  one  end  of 
its  tube,  that  end  is  the  higher.     Loose  the  screw  un- 
der the  lower  end,  and  tighten  the  one  at   the  higher 
end  till  the  bubble   is  brought   half-way  back.     Level 
the    plate    again,    and    repeat    the    operation    till    the 
bubble  will    remain   in    the    center   during    an    entire 
revolution  of  the  compass. 

2.  To  adjust  the   sights,—  Observe   through   the   slits 
a  fine  thread  made  plumb  by  a  weight.     If  both  sights 
do   not  exactly  range  with   the   thread,  file  a  little  off 
the  under  surface  of  the  highest  side. 

3.  To  adjust  the  needle. —  Bring'the  eye  nearly  in  the 
same   plane  with   the    graduated   circle,   move  with    a 
splinter  one  end  of  the  needle  to  any  division  of  the 
circle,  and  observe  whether  the  other  end  corresponds 
with  the  division  180°  from  the  first;  if  so,  the  needle 
is  said  to  cut  opposite  degrees;   if  not,  bend  the  center 
pin  with  a  small  wrench  about  one-eighth  of  an  inch 
below  the    point,  till   the   ends  of   the   needle   cut  op- 
posite   degrees.     Hold   the    needle   in   the    same   direc- 
tion, turn  the  compass  half-way  round,  and  again  see 


INSTRUMENTS.  1'91 

whether  the  needle  cuts  opposite  degrees;  if  not,  cor- 
rect half  the  error  by  bending  the  needle,  and  the  re- 
mainder by  bending  the  center  pin,  and  repeat  the 
operation  till  perfect  reversion  is  secured  in  the  first 
position. 

Try  the  needle  in  another  quarter,  and  correct  by 
bending  the  center  pin  only,  since  the  needle  was. 
straightened  by  the  previous  operation,  and  repeat  the 
operation  in  different  quarters. 

The  adjustments  are  made  by  the  maker  of  the  in- 
strument, but  the  instrument  can  be  re-adjusted  by  the 
surveyor  when  necessary. 

230.   Nature  of  the  Vernier. 

Let  the  arc  or  limb 
AB,  on  the  main  plate 
of  the  instrument,  be 
graduated  to  one  rhalf 
degrees  or  30',  numbered  each  way  from  0  at  the  mid- 
dle; and  let  the  vernier  CD,  attached  to  the  compass 
box,  which  is  movable  around  the  main  plate,  be  so 
graduated  that  30  spaces  of  the  vernier  shall  be  equal 
to  31  spaces  of  the  limb,  that  is,  equal  to  31  X  30'; 
then  1  space  of  the  vernier  will  be  equal  to  31',  and 
the  difference  between  one  space  of  the  vernier  and 
one  space  of  the  limb  will  be  31'—  30'=  1'. 

The  vernier  is  numbered  in  two  series:  the  lower, 
nearer  the  spectator,  who  is  supposed  to  stand  at  the 
south  end  of  the  instrument,  is  numbered  5,  10,  15, 
each  way  from  0;  the  upper  series  has  30  above  the 
0,  from  the  observer,  and  20  each  way  above  the  10  of 
the  lower  series. 

Let,  now,  the  0  points  of  the  vernier  and  limb  co- 
incide; then,  if  the  vernier  be  moved  forward  V  to 


192  SURVEYING. 

the  right,  which  is  done  by  means  of  a  tangent  screw, 
the  first  division  line  of  the  vernier  at  the  left  of  its 
0  will  coincide  with  the  first  division  line  of  the 
limb  at  the  left  of  its  0;  if  the  vernier  be  moved  for- 
ward 2'  to  the  right,  then  the  second  division  line  of 
the  vernier  at  the  left  of  its  0  will  coincide  with 
the  second  division  line  of  the  limb  at  the  left  of 
its  0. 

If  the  vernier  be  moved  to  the  right  so  that  its 
fifteenth  division  line  at  the  left  of  its  0  shall  coin- 
cide with  the  fifteenth  division  line  of  the  limb  at  the 
left  of  its  0,  the  vernier  will  have  moved  forward  15'. 

If  the  vernier  be  moved  more  than  15',  the  excess 
over  15'  is  found  by  reading  the  division  line,  in  the 
vernier,  which  coincides  with  a  division  line  of  the 
limb,  from  the  upper  row  of  figures  on  the  vernier, 
on  the  other  side  of  0,  and  so  on,  up  to  30',  when  the 
0  of  the  vernier  will  coincide  with  the  first  division 
line  from  the  0  of  the  limb. 

If  the  vernier  is  moved  more  than  30',  the  excess 
over  30',  up  to  15'  and  then  to  30'  is  found  as  before. 

If  the  0  of  the  vernier  coincides  with  a  division 
line  of  the  limb,  the  reading  of  the  division  line  of 
the  limb  will  be  the  true  reading. 

If  the  0  of  the  vernier  has  passed  one  or  more 
division  lines  of  the  limb,  and  does  not  coincide  with 
any,  read  the  limb  from  its  0  point  up  to  its  divis- 
ion next  preceding  the  0  of  the  vernier;  to  this  add 
the  reading  of  the  vernier,  and  the  sum  will  be  the 
true  reading. 

If  the  vernier  be  moved  to  the  left,  the  minutes  must 
be  read  off  on  the  vernier  scale  to  the  right. 

Sometimes  the  spaces  of  the  vernier  are  less  than 
the  spaces  of  the  limb;  then  if  the  vernier  be  moved 


INSTRUMENTS.  193 

either  way,  the  imitates  must  be  read  off  the  same 
way  from  the  0  of  the  vernier.  Verniers  may  be  so 
graduated  as  to  read  to  any  appreciable  angle;  but 
the  graduation  which  reads  to  minutes  is  the  most 
common. 

237.   Uses  of  the  Vernier. 

1.  To  turn  off  the  variation.  —  Let  the  instrument  be 
placed  on  some  definite  line  of  an  old  survey,  and  the 
tangent  screw  be  turned  till  the   needle  indicates  the 
same  bearing  for   the   line  as   that   given   in  the   field 
notes  of   the  original  survey. 

Then  will  the  reading  of  the  limb  and  vernier  indi- 
cate the  variation. 

2.  To  retrace  an  old  survey.  —  Turn  off  the  variation 
as  above,  and  screw  up  the  clamping  nut  beneath,  then 
old  lines  can  be  retraced  from  the  original  notes  with- 
out further  change  of  the  vernier. 

3.  To  run  a  true  meridian.  —  The   absolute  variation 
of    the    needle    being    known,    not    simply   its    change 
since    a    given    date,   move    the    vernier    to    the    right 
or    left,    according    as    the    variation    is   west    or    east, 
till   the    given  variation    is   turned   off,  screw  up    the 
clamping  nut  beneath,  and  turn  the  compass  till  the 
needle   is    made    to   cut    zeros,    then   will  'the    line   of 
sights  indicate  a  true  meridian. 

Such  a  change  in  the  position  of  the  vernier  is 
necessary  in  subdividing  the  public  lands,  after  the 
principal  lines  have  been  truly  run  with  the  solar 
compass. 

4.  To  read  the  needle  to  minutes.  —  Note  the  degrees 
given  by  the  needle,  then  turn  back  the  compass  circle, 
with   the   tangent   screw,  till  the  nearest  whole  degree 
mark  coincides  with  the  point  of  the  needle;  the  space 

S.  N.  17. 


194  SURVEYING. 

passed  over  by  the  vernier  will  be  the  minutes  which, 
added  to  the  degrees,  will  give  the  reading  of  the 
needle  to  minutes. 

This  operation  is  simplified  when  the  0  of  the  ver- 
nier is  first  made  to  coincide  with  the  0  of  the  limb ; 
otherwise  the  difference  of  the  two  readings  of  the 
vernier  must  be  taken. 

238.   Uses  of  the  Compass. 

1.  To  take  the  bearing  of  a  line.  —  Place  the  compass 
on  the  line,  turn  the  north  end  in  the  direction  of  the 
course,  and,  standing  at  the  south  end,  direct  the  sights 
to  some  well-defined  object,  as  a  flag-staff,  in  the  course. 
Read  the  bearing  from  the  north  end  of  the  needle, 
which  can  be  clone  accurately  to  quarter-degrees  by 
observing  the  position  of  the  point  of  the  needle,  since 
the  compass  circle  is  divided  into  half-degrees. 

It  will  be  observed  that  the  letters  E  and  IF,  on  the 
face  of  the  compass,  are  reversed  from  their  true  posi- 
tion. This  is  as  it  should  be;  for  if  the  sights  are 
turned  toward  the  west,  the  north  end  of  the  needle 
is  turned  toward  the  letter  W.  If  the  north  end  of 
the  needle  is  turned  toward  E,  the  sights  will  be 
turned  toward  the  east.  If  the  north  end  of  the 
needle  point  exactly  to  cither  letter  E  or  IF,  the  sights 
will  range  east  or  west. 

In  general,  to  guard  against  error,  let  the  surveyor 
turn  the  letter  ,f>  toward  himself,  and  read  the  arc  cut 
off  by  the  north  end  of  the  needle  from  the  nearest 
zero  of  the  compass  circle.  If,  for  example,  the  near- 
est 0  is  at  5,  and  the  north  end  of  the  needle  is 
turned  toward  £*,  cutting  off  25°  from  this  0,  then 
the  course  is  S  25°  E 


INSTRUMENTS.  195 

If  it  is  desired  to  find  the  bearing  to  minutes,  the 
vernier  must  be  used. 

2.  To  run  from   a  given  point  a  line  having  a  given 
bearing.  —  Place  the  compass  over  the  point,  and  turn 
it  so  that  the  reading  of  the  needle  shall  be  the  given 
bearing;    the   line   of  sights  observed   from    the   south 
end  of  the  compass  will  be  the  required  line. 

3.  To  take  angles  of  elevation,  —  Level   the  compass, 
bring  the  south   end  toward  you,  place  the  eye  at  the 
eye-piece  on   the   right   side  of   the   south    sight,   and, 
with  the   hand,  fix  a  card  on  the  front  surface  of  the 
north   sight,  so   that    its    top    edge    shall    be    at    right 
angles    to    the    divided    edge    and    coincide   with    the 
zero   mark ;    then,  sighting   over   the   top  of  the   card, 
note   upon   a   flag-staff  the   height  cut   by  the  line  of 
sight,  move  the  staff  up  the   elevation,  and  carry  the 
card  along  the  sight  until  the  line  of  sight  again  cuts 
the  same  height  on  the  staff,  read  off  the  degrees  and 
half-degrees    passed   over   by  the    card,  and  the   result 
will  be  the  angle  required. 

4.  To  take  angles  of  depression.  —  Proceed  in  the  same 
manner,  using  the  eye-piece  and  scale  on  the  opposite 
sides  of  the   sights,  and   reading   from   the  top  of  the 
standard. 

239.    Surveyor's  Transit. 

The  Surveyor's  transit  exhibited  in  the  drawing  on 
page  197  is,  in  fact,  a  transit  theodolite,  combining  the 
advantages  of  the  ordinary  transit  and  the  theodolite. 

The  vernier  plate,  carrying  two  horizontal  verniers, 
two  spirit  levels  at  right  angles,  the  telescope  and 
attachments,  moves  around  a  circle  graduated  to  half- 
degrees,  so  that,  by  the  vernier,  horizontal  angles  can 
be  taken  to  minutes,  and  any  variation  turned  off. 


196  SURVEYING. 

The  telescope  and  its  attachments,  the  clamp  and 
tangent,  the  vertical  circle,  the  level,  and  the  sights, 
give  to  this  instrument  a  great  advantage  over  the 
ordinary  compass. 

The  cross  wires,  two  fine  fibers  of  spider's  web,  ex- 
tending across  the  tube  at  right  angles,  intersect  in  a 
point  which,  when  the  wires  are  adjusted,  determines' 
the  optical  axis  or  line  of  collimutioii  of  the  telescope, 
and  enables  the  surveyor  to  fix  it  upon  an  object  with 
great  precision. 

The  clamp  and  tangent  screw  consist  of  a  ring  en- 
circling the  axis  of  the  telescope,  having  two  project- 
ing arms  —  the  one  above,  slit  through  the  middle,  hold- 
ing the  clamp  screw ;  the  other,  longer,  connected  be- 
low with  the  tangent  screw. 

The  ring  is  brought  firmly  around  the  axis  by  means 
of  the  clamp  screw,  and  the  telescope  can  be  moved  up 
or  down  by  turning  the  tangent  screw. 

The  vertical  circle,  graduated  to  half-degrees,  is  at- 
tached to  the  axis  of  the  telescope,  and,  in  connection 
with  the  vernier,  gives  the  means  of  measuring  ver- 
tical angles  to  minutes  with  great  facility. 

The  level  attached  to  the  telescope  enables  the  sur- 
veyor to  run  horizontal  lines,  or  to  find  the  difference 
of  level  between  two  points. 

Sights  on  the  telescope  are  useful  in  taking  back- 
sights without  turning  the  telescope,  and  in  sighting 
through  bushes  or  woods. 

Sights  for  right  angles  attached  to  the  plate  of  the 
instrument,  or  to  the  standards  supporting  the  tele- 
scope, afford  the  means  of  laying  off  right  angles,  or 
running  out  offsets  without  changing  the  position  of 
the  instrument. 


SURVEYOR'S  TRANSIT. 


(197) 


198  SURVEYING. 

240.    Adjustments. 

1.  The   levels   are   adjusted   in   the    same    manner   as 
those  of  the  compass,  and  when  adjusted  should  keep 
their  position  if  the  two  plates  are   clamped   together 
and  turned  on  a  common  socket. 

2.  The  needle  is  adjusted  as  in  the  compass. 

3.  The  line  of  the  collimation  is  adjusted  by  bringing 
the  intersection  of  the  wires   into  the  optical  axis  of 
the  telescope,  which  is  accomplished  as  follows  : 

Set  the  instrument  firmly  on  the  ground  and  level 
it  carefully,  then,  having  brought  the  wires  into  the 
focus  of  the  eye-piece,  adjust  the  object  glass  on  some 
well  defined  object,  as  the  edge  of  a  chimney,  at  a 
distance  of  from  two  to  five  hundred  feet.  Determine 
whether  the  vertical  wire  is  plumb  by  clamping  the 
instrument  firmly  to  the  spindle,  and  applying  the 
wire  to  the  vertical  edge  of  a  building,  or  observing 
if  it  will  move  parallel  to  a  point  a  little  -to  one  side; 
if  it  does  not,  loosen  the  cross-wire  screws,  and,  by  the 
pressure  of  the  hand  on  the  head  outside  the  tube? 
move  the  ring  within  the  tube,  to  which  the  wires 
are  attached,  gently  around  till  the  error  is  corrected. 

The  wires  being  thus  made  respectively  horizontal 
and  vertical,  fix  their-  point  of  intersection  on  the 
object  selected,  clamp  the  instrument  to  the  spindle, 
and,  having  revolved  the  telescope,  find  or  place  some 
object  in  the  opposite  direction,  at  about  the  same 
distance  from  the  instrument  as  the  first  object. 

Great  care  should  be  taken  in  turning  the  telescope 
not  to  disturb  the  position  of  the  instrument  upon  the 
spindle. 

Having  found  an  object  which  the  vertical  wire  bi- 
sects, unclamp  the  instrument,  turn  it  half-way  round, 


INSTRUMENTS.  199 

and  direct  the  telescope  to  the  first  ohject  selected, 
and  having  bisected  this  with  the  wires,  again  clamp 
the  instrument,  revolve  the  telescope  and  note  if  the 
vertical  wire  bisects  the  second  object  observed;  if 
so,  the  wires  are  adjusted,  and  the  points  bisected 
are,  with  the  center  of  the  instrument,  in  the  same 
straight  line. 

If  the  vertical  wire  does  not  bisect  the  second  point, 
ithe  space  which  separates  this  wire  from  that  point  is 
double  the  distance  of  that  point  from  a  straight  line 
drawn  through  the  first  point  and  the  center  of  the 
instrument,  as  is  shown  thus: 


Let  A  represent  the  center  of  the  instrument,  BC 
the  line  on  whose  extremities,  B  and  (7,  the  line  of 
collimation  is  to  be  adjusted,  B  the  first  object,  and  D 
the  point  which  the  wires  bisected  after  the  telescope 
was  made  to  revolve  on  its  axis.  The  side  of  the 
telescope  which  was  up  when  the  object  glass  was  di- 
rected to  B,  is  down  when  the  object  glass  is  turned 
toward  D.  When  the  telescope  is  undamped  from  its 
spindle  and  turned  half-way  round  its  vertical  axis, 
and  again  directed  to  £,  the  side  of  its  tube  which 
was  down  when  the  object  glass  was  first  directed  to 
B  will  now  be  up.  Then  clamping  the  instrument, 
and  revolving  the  telescope  about  its  axis,  and  di- 
recting it  toward  D,  the  side  of  its  tube  which  was 
down  when  the  object  glass  was  first  turned  toward 
D  will  now  be  up,  or  the  telescope  will  virtually  have 
revolved  about  its  optical  axis,  and  the  vertical  wire 
will  appear  at  E  as  far  on  one  side  of  C  as  D  is  on 
the  other  side. 


200  SUJZVEYIXG. 

To  move  the  vertical  wire  to  its  true  position,  turn 
the  capstan  head  screws  on  the  sides  of  the  telescope, 
remembering  that  the  eye -piece  inverts  the  position 
of  the  wire,  and,  therefore,  that  in  loosening  one  of 
the  screws  and  in  tightening  the  other  the  operator 
must  proceed  as  if  to  increase  the  error.  Having 
moved  back  the  vertical  wire,  as  nearly  as  can  be 
judged,  so  as  to  bisect  the  space  ED,  unclamp  the  in- 
strument, direct  the  telescope  as  at  first,  so  that  the 
cross  wires  bisect  5,  proceed  as  before,  and  continue 
the  operation  till  the  two  points  D  and  E  coincide 
at  C. 

4.  The  standards  must  be  of  the  same  height,  in 
order  that  the  wires  may  trace  a  vertical  line  when 
the  telescope  is  turned  up  or  down.  To  ascertain  this, 
and  to  make  the  correction,  proceed  as  follows : 

Having  the  line  of  collimation  previously  adjusted, 
set  the  instrument  in  a  position  where  points  of  ob- 
servation, such  as  the  point  and  base  of  a  lofty  spire, 
can  be  selected,  giving  a  long  range  in  a  vertical 
direction. 

Level  the  instrument,  fix  the  wires  on  the  top  of 
the  object,  and  clamp  to  the  spindle;  •  then  bring  the 
telescope  down  till  the  wires  bisect  some  good  point, 
either  found  or  marked  at  the  base;  turn  the  instru- 
ment half  around,  fix  the  wires  on  the  lower  point, 
clamp  to  the  spindle,  and  raise  the  telescope  to  the 
highest  object,  and  if  the  wires  bisect  it,  the  vertical 
adjustment  is  effected. 

If  the  wires  are  thrown  to  one  side,  the  standard 
opposite  that  side  is  higher  than  the  other. 

The  correction  is  made  by  turning  a  screw  under- 
neath the  sliding  piece  of  the  bearing  of  the  movable 
axis. 


INSTRUMENTS.  201 

e5.  The  vertical  circle  is  adjusted  thus :  First  care- 
fully level  the  instrument,  bring  the  zeros  of  the 
wheel  and  vernier  into  line,  and  find  or  place  some 
well  defined  point  which  is  cut  by  the  horizontal  wire; 
then  turn  the  instrument  half-way  around,  revolve 
the  telescope,  fix  the  wire  on  the  same  point  as  be- 
fore, note  if  the  zeros  are  again  in  line. 

If  not,  loosen  the  screws,  move  the  zero  over  half 
the  error,  and  again  bring  the  zeros  into  coincidence, 
and  proceed  as  before  till  the  error  is  corrected. 

6.  The  level  on  the  telescope  can  be  adjusted  thus : 
First  level  the  instrument  carefully,  and  with  the 
clamp  and  tangent  movement  to  the  axis  make  the 
telescope  horizontal  as  nearly  as  possible  with  the 
eye.  Then,  having  the  line  of  collimation  previously 
adjusted,  drive  a  stake  at  a  distance  of  from  one  to 
two  hundred  feet,  and  note  the  height  cut  by  the 
horizontal  wire  upon  a  staff  set  on  the  top  of  the 
stake. 

Fix  another  stake  in  the  opposite  direction,  at  the 
same  distance  from  the  instrument,  and,  without  dis- 
turbing the  telescope,  turn  the  instrument  upon  its 
spindle,  set  the  staff  upon  the  stake  and  drive  in  the 
ground  till  the  same  height  is  indicated  as  in  the 
first  observation. 

The  top  of  the  two  stakes  will  then  be  in  the  same 
horizontal  line,  whether  the  telescope  is  level  or  not. 

Now  remove  the  instrument  to  a  point  on  the  same 
side  of  both  stakes,  in  a  line  with  them,  and  from 
fifty  to  one  hundred  feet  from  the  nearest  one ;  again 
level  the  instrument,  clamp  the  telescope  as  nearly 
horizontal  as  possible,  and  note  the  heights  indicated 
on  the  staff  placed  first  on  the  nearest,  then  on  the 
more  distant  stake. 


202  SURVEYING. 

If  both  agree,  the  telescope  is  level ;  if  they  do  not 
agree,  then  with  the  tangent  screw  move  the  wire 
over  nearly  the  whole  error,  as  shown  at  the  distant 
stake,  and  repeat  the  operation  just  described  till  the 
horizontal  wire  will  indicate  the  same  height  at  both 
stakes,  when  the  telescope  will  be  level.  Bring  the 
bubble  into  the  center  by  the  leveling  nuts  at  the 
lend,  taking  care  not  to  disturb  the  position  of  the 
telescope,  and  the  adjustment  will  be  completed. 

The  adjustments  above  described  are  always  made 
by  the  maker  of  the  instrument,  but  the  instrument 
may  need  re-adjusting. 

241.   Uses  of  the  Transit. 

1.  The  transit  may  be   used  for  all  the  purposes  for 
which   the  compass  is  employed,  and,  in  general,  with 
much  greater  precision. 

2.  Horizontal  angles  can  be  taken   by  the   needle,  or 
without  reference  to  the  needle,  as  follows :    Level  the 
plate,  set  the  limb  at  zero,  direct  the  telescope  so  that 
the   intersection   of   the  wires   shall   fall   upon   one   of 
the  objects   selected,  clamp   the    instrument    firmly  to 
the   spindle,  unclamp   the  vernier   plate,  turn    it  with 
the  hand   till   the   intersection  of  the  wires   is   nearly 
upon  the  second  object;    then  clamp  to  the  limb,  and 
with    the    tangent    screw    fix    the    intersection    of   the 
wires   precisely  upon   the   second  object.     The  reading 
of   the  vernier  will  give  the  angle  whose  vertex  is  at 
the   center   of   the    instrument,   and   whose    sides    pass 
through  the  objects  respectively. 

3.  Vertical  angles  can  be  measured  thus:    Level  the 
instrument,    fix    the    zeros   of   the   vertical    circle   and 
vernier  in  a  line,  note   the   height  cut  upon  the   staff 


INSTRUMENTS.  203 

by  the  horizontal  wire,  carry  the  staff  up  the  eleva- 
tion or  down  the  depression,  fix  the  wire  again  upon 
the  same  point,  and  the  angle  will  be  read  off  by  the 
vernier.  Sometimes,  of  course,  it  will  be  impossible 
to  carry  the  staff  up  the  elevation,  as  in  taking  the 
angle  of  elevation  of  the  top  of  a  steeple  from  a  given 
point  in  a  horizontal  plane. 

4.  Horizontal  lines  can  be  run,  or  the  difference  of 
level  easily  found,  by  means  of  the  level  attached  to 
the  telescope. 


242.   The  Solar  Compass. 

Burt's  solar  compass,  represented  in  the  drawing  on 
page  205,  includes  the  essential  parts  of  the  magnetic 
compass,  together  with  the  solar  apparatus,  which  con- 
sists mainly  of  three  arcs  of  circles  by  which  the 
latitude  of  the  place,  the  declination  of  the  sun,  and 
the  hour  of  the  day  can  be  set  off. 

The  latitude  arc,  a,  graduated  to  quarter-degrees  and 
read  to  minutes  by  a  vernier,  has  its  center  of  motion 
in  two  pivots,  one  of  which  is  seen  at  r/,  and  is  moved 
by  the  tangent  screw,  /,  up  or  down  a  fixed  arc  of 
similar  curvature  through  a  range  of  about  35°. 

The  decimation  arc,  fo,  having  a  range  of  about  24°, 
is  graduated  to  quarter-degrees  and  read  to  minutes  by 
the  vernier,  r,  fixed  to  the  movable  arm,  /<,  which  has 
its  center  of  motion  in  the  center  of  the  declination 
arc  at  g.  The  vernier  may  be  set  to  any  reading  by 
the  tangent  screw,  k,  and  the  arm  clamped  in  any  po- 
sition by  a  screw  concealed  in  the  engraving. 

A  solar  lens,  set  in  a  rectangular  block  of  brass  at 
each  end  of  the  arm,  A,  has  its  focus  at  the  inside  of 


204  SURVEYING. 

the  opposite  block  on  the  surface  of  a  silver  plate  on 
which  are  drawn  certain  lines,  as  shown  in  the  an- 
nexed figure.  The  lines  bb,  called  hour  lines,  and  the 
lines  cc,  called  equatorial  lines,  inter- 
sect each  other  at  right  angles.  The 
rectangular  space  between  the  lines  is 
just  sufficient  to  include  the  circular 
image  of  the  sun  formed  by  the  solar  lens  on  the  op- 
posite end  of  the  arm. 

The  three  other  lines  below  the  equatorial  lines  are 
five  minutes  apart,  and  are  used  in  making  allowance 
for  refraction. 

An  equatorial  sight,  used  in  adjusting  the  solar  ap- 
paratus, is  placed  on  the  top  of  each  rectangular  block 
by  a  small  milled  head  screw,  so  as  to  be  detached  at 
pleasure. 

The  hour  arc,  c,  supported  by  the  pivots  of  the  lati- 
tude arc,  and  connected  with  that  arc  by  a  curved  arm, 
has  a  range  of  120°,  graduated  to  half-degrees  and 
figured  in  two  series,  designating  both  the  hours  and 
the  degrees;  the  middle  division  being  marked  12  and 
90  on  either  side  of  the  graduated  lines. 

The  polar  -axis,  p,  consists  of  a  hollow  socket  con- 
taining the  spindle  of  the  declination  arc,  around 
which  this  arc  can  be  moved  over  the  hour  arc, 
which  is  read  by  the  lower  edge  of  the  graduated 
side  of  the  declination  arc.  The  declination  arc  may 
be  turned  half  round,  if  required,  and  the  hour  arc 
read  by  a  point  below  y. 

The  needle  box,  /,,  with  an  arc  of  36°,  graduated  to 
half-degrees,  and  numbered  from  the  center  as  zero, 
is  attached  by  a  projecting  arm  to  a  tangent  screw,  ?, 
by  which  it  is  moved  about  its  center,  and  its  needle 


w 

3 


O 
o 

"T3 
I 


(205) 


206  SURVEYING. 

set  to  any  variation  which  may  be  read  to  minutes  by 
the  vernier  at  the  end  of  the  arm. 

The  levels  are  similar  to  those  of  the  ordinary  com- 
pass. 

Lines  of  refraction  are  drawn  on  the  inside  faces 
of  the  sights,  graduated  and  figured  to  indicate  the 
amount  allowed  for  refraction  when  the  sun  is  near 
the  horizon. 

The  adjuster  is  an  arm  used  in  adjusting  the  instru- 
ment. It  is  not  attached  to  the  instrument,  and  is 
laid  aside  in  the  box  when  the  adjustment  is  effected. 

243.    Adjustments. 

1.  The   levels   are   adjusted   by  bringing  the   bubbles 
into  the  center  of  the  tubes  by  the  leveling  screws  of 
the   tripod,  reversing   the   instrument  on   the   spindle, 
raising    or    lowering    the    ends    of   the    tubes    till    the 
bubbles  will  remain   in   the  center  during  a  complete 
revolution. 

2.  The  equatorial  lines  and  solar  lenses  are  adjusted 
as    follows :    First    detach    the   arm,  ^,  from    the    decli- 
nation  arc   by  withdrawing   the   screws   shown   in  the 
drawing   from    the    ends   of  the   posts   of   the    tangent 
screw,  kj  and    also    the    clamp    screw,  and   the   conical 
pivot  with    its    small    screws    by  which    the    arm    and 
declination  arc  are  connected. 

Attach  the  adjuster  in  the  place  of  the  arm,  /i,  by 
replacing  the  conical  pivot  and  screws,  and  insert  the 
clamp  screw  so  as  to  clamp  the  adjuster  at  any  point 
on  the  declination  arc. 

Now  level  the  instrument,  place  the  arm,  h,  on  the 
adjuster,  with  the  same  side  resting  against  the  sur- 
face of  the  declination  arc  as  before  it  was  detached, 


INSTRUMENTS.  207 

turn  the  instrument  on  its  spindle,  so  as  to  bring 
the  solar  lens  to  be  adjusted  in  the  direction  of  the 
sun,  raise  or  lower  the  adjuster  on  the  declination 
arc  till  it  can  be  clamped  in  such  a  position  as  to 
bring  the  sun's  image,  as  near  as  may  be,  between 
the  equatorial  lines  on  the  opposite  silver  plate,  and 
bring  the  image  precisely  into  position  by  the  tan- 
gent of  the  latitude  arc,  or  the  leveling  screws  of 
the  tripod.  Then  carefully  turn  the  arm  half-way 
over,  till  it  rests  upon  the  adjuster  by  the  opposite 
faces  of  the  rectangular  blocks,  and  again  observe  the 
position  of  the  sun's  image. 

If  it  remains  between  the  lines  as  before,  the  lens 
and  plate  are  in  adjustment;  if  not,  loosen  the  three 
screws  which  confine  the  plate  to  the  block,  and  move 
the  plate  under  their  heads  till  one-half  the  error  in 
the  position  of  the  sun's  image  is  removed. 

Again  bring  the  image  between  the  lines,  and  re- 
peat the  operation  till  it  will  remain  in  the  same 
situation  in  both  portions  of  the  arm,  when  the  ad- 
justment will  be  complete. 

To  adjust  the  other  lens  and  plate,  reverse  the  arm, 
end  for  end,  on  the  adjuster,  and  proceed  as  in  the 
former  case. 

Remove  the  adjuster,  and  replace  the  arm,  A,  with 
its  attachments. 

In  tightening  the  screws  over  the  silver  plate,  care 
must  be  taken  not  to  move  the  plate. 

3.  The  vernier  of  the  declination  arc  is  adjusted  as- 
follows :  Having  leveled  the  instrument,  and  turned 
its  lens  in  the  direction  of  the  sun,  clamp  to  the 
spindle,  and  set  the  vernier,  v,  of  the  declination  arc 
at  zero,  by  means  of  the  tangent  screw,  fc,  and  clamp 
to  the  arc. 


208  SURVEYING. 

See  that  the  spindle  moves  easily  and  truly  in  the 
socket,  or  polar  axis,  and  raise  or  lower  the  latitude 
arc  by  turning  the  tangent  screw,  /,  till  the  sun's  im- 
age is  brought  between  the  equatorial  lines  on  one  of 
the  plates;  clamp  the  latitude  arc  by  the  screw,  and 
bring  the  image  precisely  into  position  by  the  level- 
ing screws  of  the  tripod  or  socket,  and  without  dis- 
turbing the  instrument  carefully  revolve  the  arm,  //, 
till  the  opposite  lens  and  plate  are  brought  in  the 
direction  of  the  sun,  and  note  if  the  sun's  image 
comes  between  the  lines  as  before. 

If  the  sun's  image  comes  between  the  lines,  there  is 
no  index  error  of  the  declination  arc;  if  not,  then  with 
the  tangent  screw,  k,  move  the  arm  till  the  sun's  im- 
age passes  over  half  the  error,  and  again  bring  the 
image  between  the  lines,  and  repeat  the  operation  as 
before  till  the  image  will  occupy  the  same  position 
on  both  plates. 

We  shall  now  find  that  the  zero  marks  on  the  arc 
and  the  vernier  do  not  correspond ;  and  to  remedy 
this  error,  the  little  flat-head  screws  above  the  vernier 
must  be  loosened  till  it  can  be  moved  so  as  to  make 
the  zeros  coincide,  when  the  operation  will  be  com- 
plete. 

4.  The  solar  apparatus  is  adjusted  to  the  sights  as 
follows:  First  level  the  instrument,  then  with  the 
clamp  and  tangent  screws  set  the  main  plate  at  90° 
by  the  verniers  and  horizontal  limb.  Then  remove  the 
.clamp  screw,  and  raise  the  latitude  arc  till  the  polar 
axis  is  by  estimation  very  nearly  horizontal,  and,  if 
necessary,  tighten  the  screws  on  the  pivots  of  the  arc 
so  as  to  retain  it  in  this  position. 

Fix  the  vernier  of  the  declination  arc  at  zero,  and 
direct  the  equatorial  sights  to  some  distant  and  well- 


INSTRUMENTS.      .  209 

marked  object,  and  observe  the  same  through  the  com- 
pass sights.  If  the  same  object  is  seen  through  both, 
and  the  verniers  read  to  90°  on  the  limb,  the  adjust- 
ment is  complete ;  if  not,  the  correction  must  be  made 
by  moving  the  sights  or  changing  the  position  of  the 
verniers. 

The  adjustments  are  all  made  by  the  maker  of  the 
instrument,  and,  ordinarily,  need  not  concern  the  sur- 
veyor, as  the  instrument  is  very  little  liable  to  de- 
rangement. 

244.   Use  of  the  Solar  Compass. 

The  declination  of  the  sun,  or  its  angular  distance 
from  the  celestial  equator,  must  be  set  off  on  the 
declination  arc. 

The  declination  of  the  sun  for  apparent  noon  at 
Greenwich,  England,  is  given  from  year  to  year  in 
the  Nautical  Almanac, 

To  determine  the  declination  for  another  place  and 
hour,  allowance  must  be  made  for  the  difference  of 
time  arising  from  longitude,  and  for  the  change  of 
declination  from  hour  to  hour. 

The  longitude  of  the  place  can  be  determined  with 
sufficient  accuracy  by  reference  to  that  of  given  promi- 
nent places  which  are  situated  nearly  on  the  same 
meridian. 

The  difference  of  longitude,  divided  by  15,  will,  by 
changing  degrees,  minutes,  and  seconds  into  hours, 
minutes,  and  seconds,  give  the  difference  of  time, 
which  is  usually  taken  to  the  nearest  hour,  as  it 
will  be  sufficiently  accurate. 

In  practice,  surveyors  in  states  just  east  of  the  Mis- 
sissippi allow  a  difference  of  6  hours  for  longitude; 
S.  N.  18. 


210  SURVEYING. 

7  hours  for  about  the  longitude  of  Santa  Fe;  8  hours 
for  California  and  Oregon;  5  hours  for  the  eastern 
portions  of  the  United  States. 

Having  found  the  hour  at  any  place  from  its  longi- 
tude when  it  is  noon  at  Greenwich,  the  declination 
for  noon  at  Greenwich  will  be  the  declination  for  the 
determined  hour  at  the  given  place. 

To  find  the  declination  for  the  following  hours  of 
the  day,  add  or  subtract,  for  each  succeeding  hour,  the 
difference  of  declination  for  1  hour,  as  given  in  the 
almanac. 

Thus,  let  it  be  required  to  find  the  declination  of 
the  sun  for  the  different  hours  of  May  20th,  1873. 
W.  Ion.  95°.  95°  =  6  h.  20  m.,  practically  G  h. 

Sun's  dec.,  Greenwich,  noon  ==  20°  3'  14".6 

.'.     Sun's  dec.,  Ion.  95°,  6  A.  M.  ==  20°  3'  14".6 

Add  difference  for  1  h.  I  A1^3 

Sun's  dec.  7  A.  M.  -=  20°~  37  45".63 

Add  difference  for  1  h.  =  _3J-^03 

Sun's  dec.  8  A.  M.  ==  20°  4'  16".66 

In  like  manner  proceed  for  the  remaining  hours. 

'Such  a  calculation  should  be  made  before  beginning 
the  work  of  the  day. 

Refraction,  or  the  bending  of  the  sun's  rays  as  they 
pass  obliquely  through  the  atmosphere,  affects  its  dec- 
lination by  increasing  its  apparent  altitude. 

The  amount  of  refraction  depends  upon  the  altitude, 
being  less  as  the  altitude  is  greater.  At  the  horizon 
the  refraction  is  35';  at  the  altitude  of  45°,  1';  at 
the  zenith,  0. 

Meridional  refraction,  by  increasing  the  apparent  al- 
titude of  the  sun,  when  on  the  meridian,  increases  or 


INSTRUMENTS.  211 

diminishes  its  apparent  declination   according  as  it  is 
north  or  south  of  the  equator. 

To  find  the  amount  of  meridional  refraction,  we 
must  first  find  the  meridional  altitude  of  the  sun 
for  the  given  latitude,  which  is  equal  to  the  comple- 
ment of  the  latitude,  plus  or  minus  the  declination, 
according  as  the  sun  is  north  or  south  of  the  equator. 

The  meridional  altitude  of  the  sun  being  given, 
the  tables  will  give  the  refraction. 

The  meridional  refraction,  being  quite  small,  may 
be  disregarded  in  practice  except  when  great  accuracy 
is  required,  as  in  running  great  standard  meridians 
or  base  lines. 

Incidental  refraction,  as  affected  by  the  hour  of  the 
day  and  the  state  of  the  atmosphere,  can  not,  in  prac- 
tice, be  determined  by  a  precise  calculation. 

It  will  about  compensate  for  incidental  refraction  to 
keep  the  image  of  the  sun  square  between  the  equi- 
noctial lines  for  the  middle  of  the  ,day,  but  toward 
morning  or  evening,  to  run  the  image,  which  is  then 
hazy  round  the  edge,  so  that  the  hazy  edge  shall  over- 
lap one  or  two  lines  of  the  spaces  below. 

To  set  off  the  latitude,  find  the  declination  of  the  sun 
for  the  given  day  at  noon,  and  set  it  off  on  the  decli- 
nation arc,  and  clamp  the  arm  firmly  to  the  arc. 

Find  in  the  almanac  the  equation  of  time  for  the 
given  day,  in  order  to  ascertain  the  time  when  the 
sun  will  reach  the  meridian. 

About  twenty  minutes  before  noon,  set  up  the  in- 
strument, level  it  carefully,  fix  the  divided  surface  of 
the  declination  arc  at  12  on  the  hour  circle,  and  turn 
the  instrument  on  its  spindle  till  the  solar  lens  is 
brought  into  the  direction  of  the  sun. 


212  SURVEYING. 

Loosen  the  clamp  screw  of  the  latitude  arc,  raise 
or  lower  this  arc  with  the  tangent  screw  till  the  im- 
age of  the  sun  is  brought  precisely  between  the  equa- 
torial lines,  and  turn  the  instrument  so  as  to  keep 
the  image  between  the  hour  lines  on  the  plate. 

As  the  sun  ascends,  in  approaching  the  meridian, 
its  image  will  move  below  the  lines,  and  the  arc  must 
be  moved  to  follow  it.  Keep  the  image  between  the 
two  sets  of  lines  till  it  begins  to  pass  above  the 
equatorial,  which  is  the  moment  after  it  passes  the 
meridian. 

Read  off  the  vernier  of  the  arc,  and  we  have  the 
latitude  of  the  place  which  is  to  be  set  off  on  the 
latitude  arc. 

To  run  lines  with  the  solar  compass. — Having  adjusted 
the  instrument  and  set  off  the  declination  and  latitude, 
the  surveyor  places  the  instrument  over  the  station, 
levels  it  carefully,  clamps  the  plates  at  zero  on  the 
horizontal  limb,  and  directs  the  sights  north  and 
south,  approximately,  by  the  needle. 

The  solar  lens  is  then  turned  toward  the  sun,  and 
with  one  hand  on  the  instrument,  and  the  other 
on  the  revolving  arm,  both  are  moved  from  side  to 
side  till  the  image  of  the  sun  is  made  to  appear  on 
the  silver  plate,  and  is  brought  precisely  within  the 
equatorial  lines,  when  the  line  of  sights  will  indicate  the 
true  meridian. 

In  running  an  east  and  west  line,  the  verniers  of 
the  horizontal  limb  are  set  at  90°,  and  the  sun's  im- 
age kept  between  the  equatorial  lines. 

The  needle  is  made  to  indicate  zero  on  the  arc  of 
the  compass  box  by  turning  the  tangent  screw.  Lines 
can  then  be  run  by  the  needle  in  the  temporary  dis- 
appearance of  the  sun. 


INSTRUMENTS.  213 

The  variation  of  the  needle,  which  should  be  noted 
at  every  station,  is  read  off  to  minutes  on  the  arc 
along  the  edge  of  which  the  vernier  of  the  needle 
box  moves. 

Since  the  limb  must  be  clamped  at  0  when  the  sun's 
image  is  in  position,  in  order  that  the  sights  may  indi- 
cate the  meridian,  it  is  evident  that  the  bearing  of 
any  line  may  be  found  by  the  solar  compass,  as  well  as 
by  the  compass  or  transit. 

In  running  long  lines,  allowance  must  be  made  for 
the  curvature  of  the  earth.  Thus,  in  running  north 
or  south  the  latitude  changes  V  for  92.30  ch.;  and  six 
miles,  or  one  side  of  a  township,  requires  a  change  of 
5'  12"  on  the  latitude  arc. 

In  running  east  and  west  lines,  the  sights  are  set 
at  90°  on  the  limb,  and  the  line  run  at  right  angles 
to  the  meridian ;  but  this  line,  if  sufficiently  produced, 
would  cross  the  equator.  Hence,  at  the  next  station, 
a  backsight  is  taken,  and  one-half  the  error  is  set  off 
for  the  next  foresight  on  the  side  toward  the  pole. 

The  most  favorable  season  for  using  the  solar  com- 
pass is  the  summer;  and  the  most  favorable  time  of 
day,  between  8  and  11  A.  M.,  and  1  and  5  P.  M. 

A  solar  telescope  compass  is  sometimes  used;  and, 
in  this  case,  the  telescope  is  placed  at  one  side  of  the 
center.  All  error  from  this  position  of  the  telescope  is 
avoided  by  an  offset  from  the  flag-staff. 

The  solar  compass,  while  indispensable  in  the  survey 
of  public  lands,  can  be  used,  in  common  practice,  with 
considerable  advantage  over  ordinary  needle  instru- 
ments, since  lines  can  be  run  by  it  without  regard  to 
the  variation  of  the  needle  or  local  attraction,  and  the 
bearings  being  taken  from  the  true  meridian  will  re- 
main constant  for  all  time. 


214 


SURVEYING. 


245.    Dividers  and  Pens. 


1.  Dividers  with  lead-pencil. 

2.  Hair  dividers  with  one  leg  movable  by  screw. 

a,  6.  Lengthening  bar  and  pen  which  may  be  inserted 
together  or  the  pen  alone  instead  of  pencil  leg. 

3.  Bow  pen  with  spring  and  adjusting  screw. 

4.  Spacing  dividers. 

5.  Drawing  pen. 

246.   Parallel  Rulers. 


1.  Parallel  ruler  for  drawing  parallel  lines. 

2.  Sliding  parallel  ruler  with  scales. 


INSTRUMENTS. 


215 


247.    Diagonal  Scale. 


Let  c?e  be  .1,  then  the  distance  from  ad  to  ae  on  the 
first  line  above  ab  is  .01,  on  the  second  line  .02,  etc. 

Let  it  be  required  to  lay  off  on  AB  4.63. 

Place  one  foot  of  the  dividers  at  the  intersection  of 
the  diagonal  line,  6,  and  the  horizontal  line,  3.  Extend 
the  other  foot  till  the  horizontal  line,  3,  intersects  the 
vertical  line,  4,  then  will  the  distance  from  one  point 
of  the  dividers  to  the  other  be  4.63. 

Now  place  one  foot  of  the  dividers  at  A,  and  the 
other  at  B,  then  AB  will  be  4.63. 


248.   Protractors. 


These  protractors  are  used  in  laying  off  or  measur- 
ing angles.  The  vertex  of  the  angle  is  at  the  center, 
and  one  side  is  made  to  coincide  with  the  horizontal 
line  passing  through  the  center;  then,  counting  the 
degrees,  from  the  horizontal  line  round  the  circumfer- 
ence till  the  required  degree  is  reached,  and  drawing 


216"  SURVEYING. 

a   line   from   this   degree    to   the  center,  we  shall  have 
the  angle  required. 

The  first  of  these  protractors  will  give  angles  to 
quarter-degrees ;  and  the  second,  by  means  of  a  ver- 
nier, to  8'. 

Instruments  may  be  multiplied  indefinitely,  but  the 
manner  of  using  them  will  be  readily  discovered  by 
the  ingenious  operator. 


SURVEY   OF   PUBLIC  LANDS. 
249.    Division  into  Townships. 

In  the  rectangular  system  of  surveying  the  public 
lands,  adopted  by  the  government,  two  principal  lines 
—an  east  and  west  line,  called  a  base  line,  and  a  north 
and  south  line,  called  a  principal  meridian  —  are  estab- 
lished before  the  survey  of  the  townships. 

Six  miles  to  the  north  of  the  base  line  another  east 
and  west  line  is  run,  and  six  miles  to  the  north  of  this 
another,  and  so  on. 

Every  fifth  parallel  from  the  base  is  called  a  standard 
parallel,  or  correction  line. 

Six  miles  to  the  west  of  the  principal  meridian, 
measured  on  the  base  line,  another  north  and  south 
line  is  run  to  the  first  standard  parallel,  and  six 
miles  to  the  west  of  this  another,  and  so  on. 

The  intersection  of  the  east  and  west  with  the  north 
and  south  lines  divides  the  tract  into  townships,  which 
would  be  exactly  six  miles  square  were  it  not  for  the 
convergence  of  the  meridians. 

To  preserve  as  nearly  as  possible  the  form  and  size 
of  the  townships,  the  standard  parallels  before  men- 


PUBLIC  LANDS.  217 

tioned   are   established,  which   serve   as   hase   lines   for 
the  townships  north  up  to  the  next  standard  parallel. 

Tiers  of  townships  north  and  south  arc  called  ranges, 
and  are  numbered  east  or  west,  as  the  case  may  be, 
from  the  principal  meridian. 

Lines  running  north  and  south,  bounding  the  town- 
ships on  the  east  and  west,  are  called  range  lines. 

Lines  running  east  and  west,  bounding  the  townships 
on  the  north  and  south,  are  called  township  lines. 

A  township  marked  thus,  T.  5  TV.,  R.  4  W.,  read 
township  five  north,  range  four  west,  would  be  in  the 
fifth  tier  north  of  the  base  line,  and  in  the  fourth  tier 
west  of  the  principal  meridian. 

Townships  are  divided  into  sections,  or  square  miles, 
containing  640  acres;  each  section  into  four  quarter 
sections,  each  quarter  section  into  two  half-quarter  sec- 
tions, and  each  half-quarter  section  into  two  quarter- 
quarter  sections.  These  are  called  legal  subdivisions, 
and  are  the  only  divisions  recognized  by  the  govern- 
ment, except  pieces  made  fractional  by  water-courses 
or  other  natural  agencies. 

On  base  lines  and  standard  parallels  two  sets  of 
corners  are  established. 

1.  Standard  corners,  established  when  these  lines  are 
run,  embracing    township,  section,  and   quarter-section 
corners,  common  to  two  townships,  sections,  or  quarter 
sections  north  of  the  base  line  or  standard  parallels. 

2.  Closing  corners,  established  when  exterior  and  sub- 
division lines  close  on  them  from  the  south,  embracing 
township   and   section    corners,  common    to    two    town- 
ships or  sections  south  of  the  standard  parallels. 

In  consequence  of  the  convergence  of  the  meridians, 
the  north  and  south  lines,  produced  to  the  standard 

S.  N.  19. 


218 


SURVEYING. 


parallels,  will  not  close  on  the  standard  corners  previ- 
ously established,  but  will  strike  the  standard  parallels 
to  the  east  or  west  of  the  standard  corners,  making 
the  closing  corners  east  or  west  of  the  standard  cor- 
ners, according  as  the  field  of  operation  is  west  or  east 
of  the  principal  meridian, 

The  following  diagram  will  illustrate  the  subject : 
AB  is  the  base  line. 


A' 


AC,  the  principal  me- 
ridian. 

A'B',  a  standard  paral- 
lel. 

ab,  cd,  etc.,  township 
lines. 

ijj  klj  etc.,  range  lines. 

,<?,  «.,  Wj  etc.,  standard 
corners. 

j,  I,  ?i,  etc.,  closing  cor- 
ners. 

The  distances  js,  Zit,  etc.,  are  measured  and  recorded 
in  the  field  book. 

The  details  of  running  lines  will  be  given  after 
describing  the  methods  of  perpetuating  corners,  the 
process  of  chaining,  and  the  method  of  marking  lines. 

Burt's  improved  solar  compass  is  used  in  surveying 
standard  and  township  lines,  but  the  ordinary  compass 
may  be  used  in  subdividing. 


250.    Methods  of  Perpetuating  Corners. 

1.  Corner  trees.  —  A  sound  tree,  five  inches  or  more 
in  diameter,  standing  exactly  at  a  corner,  is  the  best 
monument. 


PUBLIC  LANDS.  219 

2.  Corner  stones.  —  A  stone,  at  least   14   inches   long 
and    6    inches    square,    set    from    two-thirds    to    three- 
fourths    in    the    ground,    is    preferred    to   other    monu- 
ments, except  a  tree. 

3.  Posts  and  witness  trees. — In  the  absence  of  corner 
trees    and    stones,   when    trees    are    near,    a    post    may 
be   planted  and  witnessed  by  taking  the   bearing  and 
distance  of   two  or  more   trees   in   different   directions 
from  the  corner.    These  trees  are  marked  by  a  blaze  in 
which  is  marked  the  number  of  the  township,  range, 
and  section.     A  notch   is  cut  in  the  lower  end  of  the 
blaze,  under  which  another  blaze  is  made  in  which  are 
cut  the  letters  B.  T7.,  signifying  bearing  tree. 

4.  Posts,   mounds,   and  witness   pits. — When    neither 
corner    trees,   stones,   nor  witness    trees    are    available, 
corners  may  be  marked  by  posts,  mounds,  and  witness 
pits.     The  posts  are  planted  12  inches  in  the  ground, 
and  at  the  lower  end,  on  the  north  or  west  side,  accord- 
ing as  the  course   is  north  or  west,  a  marked  stone,  a 
small  quantity  of  charcoal,  or  a  charred  stake  must  be 
deposited.     Four    pits    are   dug,   6  feet   from   the   post, 
on  opposite  sides,  2   feet   square   and   1   foot  deep,  and 
the  excavated  earth   packed  round   the   post  within   1 
foot  of  the  top.     If  sod  is  to  be  had,  it  is   to  be   used 
in  covering  the  mounds. 

The   method   of   marking   the  corner   is  to  be   care- 
fully noted  in  the  field  book. 


251.   Township  Corners. 

1.  Posts  used  in  marking  township  corners  must  be 
4  feet  in  length,  and  5  inches,  at  least,  in  diameter. 
These  posts  are  to  be  set  2  feet  in  the  ground,  and 


220  SURVEYIXG. 

the   upper   part   squared   to   receive   the   marks   to   be 
cut  on  them. 


T.  2  X. 


P.  31V. 
S.  31. 


T.  1  3f. 


P.  4  W. 
S.I. 


T.  IN. 
E .  3  AN". 


S.  6. 


If  the  corner  is  common 
to  four  townships,  the  post 
is  set  so  as  to  present  the 
angles  in  the  direction  of 
the  line;  and  the  number 
of  the  township,  range,  and 
section  must  be  marked  on 
the  side  facing,  and  six 
notches  cut  on  each  of  the 
four  edges. 

If  the  township  corner  is  on  a  base  line  or  standard 
parallel,  unless  it  is  also  on  the  principal  meridian,  it 
will  be  common  to  two  townships  only;  and  if  these 
are  on  the  north,  the  corner  will  be  a  standard  corner. 
In  this  case,  six  notches  are  cut  on  the  east,  north, 
and  west  edges,  but  not  on  the  south  edge,  and  the 
letters  S.  (7.,  signifying  standard  corner,  cut  on  the 
flat  surface. 

If  the  corner  is  common  to  two  townships  on  the 
south,  but  not  on  the  north,  it  will  be  a  closing  cor- 
ner, and  six  notches  are  cut  on  the  east,  south,  and 
west  edges,  but  not  oh  the  north  edge,  and  the  let- 
ters C.  C.,  signifying  closing  corner,  cut  on  the  flat 
surface. 

2.  Township  corner  stones  should  be  inserted  at  least 
10  inches  in  the  ground,  with  their  sides  facing  the 
cardinal  points  of  the  compass,  and  small  mounds  of 
stones  heaped  against  them. 

These  corner  stones  are  notched  in  the  same  manner 
as  posts  in  similar  circumstances,  but  are  not  otherwise 
marked. 


PUBLIC  LANDS.  221 

3.  A  tree  of  proper  size  on  the  corner  is  marked  in 
the  same  manner  as  a  post. 

The  mounds,  when  made  round  the  posts,  must  be  5 
feet  in  diameter  at  the  base,  and  2J  feet  high.  The 
posts,  therefore,  must  be  4J  feet  long,  so  as  to  be  1  foot 
in  the  ground  and  1  foot  above  the  top  of  the  mound. 

Witness  pits  for  township  corners  must  be  2  feet 
long,  1^  feet  wide,  and  1  foot  deep.  If  the  corner  is 
common  to  four  townships,  there  will  be  four  pits 
placed  lengthwise  on  the  lines ;  but  if  the  corner  is 
common  to  only  two  townships,  only  three  pits  are 
dug,  and  are  placed  lengthwise  on  the  lines.  Thus  the 
kind  of  township  corners  are  readily  distinguished. 

These  pits  are  made  only  in  the  absence  of  witness 
trees,  which  are  to  be  selected,  if  possible,  one  from 
each  township. 

252.    Section  Corners. 

Section  corners  are  established  at  intervals  ,of  80 
chains  or  1  mile,  and  are  perpetuated  by  the  follow- 
ing methods : 

1.  Section  corner  posts  are  4  feet  in  length,  and   at 

least  4  inches   in   diameter.     They  are  planted   2   feet 

in  the  ground,  and  the  part  above  the  ground  squared 
to  receive  the  marks. 

If  the  corner  is  common  to  four  sections,  the  post  is 
set  cornerwise  to  the  lines,  the  number  of  the  section 
is  marked  on  the  side  facing  it,  and  the  number  of  the 
township  and  range  on  the  north-east  face. 

Mile-posts  on  township  lines  have  as  many  notches 
on  the  corresponding  edges  as  they  are  miles  from  the 
respective  township  corners. 


222  SURVEYING. 

Section  posts  within  a  township  have  as  many 
notches  on  the  south  and  east  edges  as  they  are  miles 
from  the  south  and  east  boundaries  of  the  township; 
but  no  notches  are  cut  on  the  north  and  west  edges. 

Section  posts  must  be  witnessed  by  trees,  one  in 
each  section,  or,  in  the  absence  of  trees,  by  pits  18 
inches  square  and  12  inches  deep. 

2.  Section  corner  mounds  are  4|  feet   in  diameter  at 
the   base,  and   2  feet   high.     The   post   must   be  4  feet 
long,  1  foot  in  the  ground,  and  1  foot  high  above  the 
mound,  and  at  least  3  inches  square. 

At  corners  common  to  four  sections,  the  edges  are 
in  the  direction  of  the  cardinal  points;  but  at  cor- 
ners common  only  to  two  sections,  the  flattened  sides 
face  the  cardinal  points. 

Section  posts  in  mounds  are  to  be  marked  and  wit- 
nessed in  the  same  manner  as  the  post  without  the 
mound. 

3.  Stones  used  to  mark   section  corners  on  township 
lines  are   set  with   their  edges  in  the  direction  of  the 
line;    but    for    interior    sections   they  face    the    north. 
They  are  witnessed  in  the  same  manner  as  posts,  but 
are  not  marked  except  by  notches. 

4.  Section  corner  trees  are  marked  and  witnessed  the 
same  as  posts. 

253.   Quarter  Section  Corners. 

Quarter  section  corners  are  established  at  intervals 
of  40  chains  or  half  a  mile,  except  in  the  north  or 
west  tiers  of  sections  of  a  township. 

In  subdividing  these  sections,  the  quarter  post  is 
placed  40  chains  from  the  interior  section  corner,  so 


PUBLIC  LANDS.  223 

that  the  excess  or  deficiency  shall  fall  in  the  last  half 
mile. 

Quarter  section  corners  are  not  required  to  be  estab- 
lished on  base  or  standard  parallel  lines  on  the  north. 

The  methods  of  perpetuating   these   corners  are   the 
following  : 

1.  Quarter  section  posts,  4  feet  in  length  and  4  inches 
in    diameter,   are    planted   or   driven    2    feet   into    the 
ground,  and   the   part   above   the   ground   squared   and 
marked  J  £,  signifying  quarter  section.    These  corners 
are  witnessed  by  two  bearing  trees. 

2.  Quarter  section  mounds  are,  like  section  mounds, 
packed  round  the  posts,  and   pits  may  be  used  in  the 
absence  of   witness  trees. 

3.  Quarter  section  stones  have  J-  cut  on  the  west  side 
of  north   and   south   lines,  and   on   the   north   side   of 
east  and  west  lines,  and  are  witnessed  by  two  bearing 
trees  or  pits. 

4.  A  quarter  section  tree  is  marked  and  witnessed  in 
the  same  manner  as  a  post. 


254.   Meander  Corners. 

Meander  corners  are  the  intersections  of  township  or 
section  lines  with  the  banks  of  lakes,  bayous,  or  navi- 
gable rivers. 

These  corners  are  marked  by  the  following  methods : 

1.  Meander   posts  of  the  same   size   as  section  posts, 
are  planted  firmly  in  the  ground,  and  witnessed  by  two 
bearing  trees  or  pits,  but  are  not  marked. 

2.  Mounds  of  the  same  size  as  those   for  section  cor- 
ners are,  in  the  absence  of  witness  trees,  formed  round 


224  SURVEYING. 

the  posts,  and  a  pit  dug  exactly  on    the   line,  8  links 
further  from  the  water  than  the  mound. 

3.  Stones  or  trees,  witnessed  in  the  same  manner  as 
posts,  may  be  employed. 

255.    Chaining. 

Eleven  tally  pins  are  employed,  ten  of  which  are 
taken  by  the  fore  chainman,  or  leader,  and  the  re- 
maining one  by  the  hind  chainman,  or  follower,  who 
sticks  it  at  the  beginning  of  the  course,  and  against  it 
brings  the  handle  at  one  end  of  the  chain. 

The  leader,  holding  the  other  handle  of  the  chain 
and  one  pin  in  his  right  hand,  draws  out  the  chain 
to  its  full  length  in  the  direction  of  the  course;  both 
taking  care  that  the  chain  is  free  from  kinks. 

The  leader  standing  to  the  left  of  the  line,  so  as  not 
to  obstruct  the  range,  with  his  right  arm  extended, 
draws  the  chain  tight,  brings  the  pin  into  line  accord- 
ing to  the  order  "right"  or  "left,"  from  the  follower, 
sticks  it  at  the  order  "down"  by  pressing  his  left 
hand  on  the  top  of  the  pin,  and  replies  "  down." 

The  follower  then  withdraws  his  pin,  and  both  ad- 
vance, the  leader  drawing  the  chain  in  the  direction 
of  the  course,  but  a  little  to  one  side  to  avoid  drag- 
ging out  the  pin,  till  the  follower  comes  up  to  the  pin, 
against  which  he  brings  the  handle  at  his  end  of  the 
chain,  and  directs  the  sticking  of  another  pin,  as  be- 
fore, and  so  on. 

When  the  leader  has  stuck  his  last  pin,  he  cries 
"tally,"  which  is  repeated  by  the  other,  and  each  regis- 
ters the  tally  by  slipping  a  ring  on  a  belt. 

The  follower  then  comes  forward,  and  counting  in 
presence  of  his  fellow,  to  avoid  mistake,  the  pins  taken 


PUBLIC  LANDS.  225 

up,  takes  the  foreward  end  of  the  chain  and  proceeds, 
as  the  leader,  for  another  tally. 

If  a  whole  chain  is  employed,  a  tally  is  ten  chains; 
and  accordingly  four  tallies  make  half  a  mile,  and  eight 
tallies  a  mile. 

If  a  half-chain  is  employed,  a  tally  is  five  chains, 
eight  tallies  are  half  a  mile,  and  sixteeen  tallies  a 
mile. 

In  measuring  up  or  down  a  hill,  the  chain  must  be 
kept  horizontal,  so  that  it  is  often  necessary  to  use 
but  a  portion  of  the  chain. 

The  chain  employed  in  the  field  must  be  compared, 
from  day  to  day,  with  a  standard  chain  furnished  by 
the  Surveyor-General,  and  any  variation  promptly  cor- 
rected. 

256.    Marking  Lines. 

Line  trees,  called  also  "station  trees,"  or  "sight  trees," 
are  marked  by  two  notches  on  each  side  of  the  tree, 
in  the  direction  of  the  line. 

The  line  is  marked,  so  as  to  be  easily  followed,  by 
blazing  a  sufficient  number  of  trees  near  the  line  on 
two  sides  quartering  toward  the  line. 

Saplings  near  the  line  are  cut  partly  off  by  a  blow 
from  the  ax,  at  the  usual  height  of  blazes,  and  bent 
at  right  angles  to  the  line. 

Random  lines  are  not  marked  by  blazing  trees;  but 
to  enable  the  surveyor  to  retrace  the  line  on  his  re- 
turn, bushes  are  lopped  and  bent  in  the  direction  of 
the  line,  and  stakes  are  driven  every  ten  chains,  which 
are  pulled  up  when  the  true  line  is  established. 

Insuperable  objects,  such  as  ponds,  marshes,  etc.,  are 
passed  by  making  right-angled  offsets,  or  by  trigono- 


226  SURVEYING. 

metrical  operations,  a  complete  record  of   which   must 
be  made  in  the  field  book. 


257.   Initial  Point  and  Principal  Lines. 

1.  The   initial   point,   which    is   usually  some    perma- 
nent natural  object,  as  the  confluence  of  two  rivers,  or 
an  isolated  mountain,  is  first  selected. 

2.  Principal  meridians  are  run  from  the  initial  points 
due  north  or  due   south,   and  the   quarter  section,  sec- 
tion, and   township   corners   on    these    lines   are  accu- 
rately located  and   perpetuated. 

The  following  are  the  principal  meridians  already 
established  : 

1st.  The  first  runs  north  from  the  mouth  of  the 
Great  Miami  river,  between  Ohio  and  Indiana,  to  the 
south  line  of  'Michigan. 

2d.  The  second  runs  north  from  the  mouth  of  the 
Little  Blue  river  through  the  center  of  Indiana  to  its 
north  line. 

3d.  The  third  runs  north  from  the  mouth  of  the 
Ohio  river  through  Illinois  to  its  north  line. 

4th.  The  fourth  runs  north  from  the  Illinois  river 
through  the  western  part  of  Illinois  and  the  center  of 
Wisconsin  to  Lake  Superior. 

5th.  The  fifth  runs  north  from  the  mouth  of  the  Ar- 
kansas river  through  the  eastern  portion  of  Arkansas, 
Missouri,  and  Iowa,  and  regulates  the  surveys  in  Min- 
nesota west  of  the  Mississippi  river,  and  the  surveys 
in  Dakota  east  of  the  Missouri  river. 

6th.  The  sixth  commences  on  the  Arkansas  river, 
in  Kansas,  and  runs  north  through  the  eastern  part 
of  Kansas  and  Nebraska  to  the  Missouri  river. 


PUBLIC  LANDS. 


227 


7th.  Independent  meridians. — These  are  the  Independent 
meridian  of  New  Mexico,  the  Salt  Lake  meridian  in  Utah, 
the  Willamette  meridian  of  Oregon  and  Washington,  and 
the  Humboldt  meridian,  the  ML  Diablo  meridian,  and  the 
St.  Bernardino  meridian  of  California. 

3.  Base  lines  are  run  from  the  initial  points  due  east 
or  due  west,  and  the  quarter  section,  section,  and  town- 
ship corners,  for  the  land   north  of  the  line,  are  accu- 
rately located,  at  full  measure,  and  perpetuated. 

4.  Standard   parallels  are   also   run  due   east  or  due 
west    thirty    miles    north    of    the    base    line   or   other 
standard  parallel,  and  the  corners  located  and  perpetu- 
ated as  on  the  base  li'ne. 

5.  Range  lines  are  run  between  the  ranges  of  town- 
ships due   north   from  a  base  line  or  standard  parallel 
to  the  next  standard  parallel. 


258.   Exterior  or  Township  Lines. 


S                                                       JM 

P' 

63 
51 

38 

87 

14 

13 

14 

13 

28 

27 

49  50 

46 

36           37 

25  26 
24 

11  12 
10 

J2            11 
10 

26           26 
24 

46  47 
45 

34           35 

22  23 
21 

8  9 

7 

9              8 
7 

23              22 
21 

43  44 
42 

32           33 
31 

19  1!0 
18 

5  6 
4 

6              5 
4 

20              1« 

18 

40  il 
39 

29           £0 

US 

Iti  IT 
15 

3 
1 

3               2 

1 

17           16 
15 

U                                                          P 

£ 

In  the  above  diagram  let  P  denote  the  initial  point, 
PM  the  principal  meridian,  BL  the  base  line,  SP'  the 


228  SURVEYING. 

first  standard  parallel  north,  and  let  the  squares  denote 
townships. 

1.  For  townships  west  of  the  meridian,  begin  at  the 
first  pre-established  township  corner  on  the  base  line 
west  of  the  meridian.  This  is  the  S.  W.  corner  of 
T.  1  N.,  R.  1  TT.,  and  is  marked  1  in  the  diagram. 

Measure  thence  due  north  480  chains,  establishing 
the  quarter  section  and  section  corners,  to  2,  at  which 
point  establish  the  corner  common  to  T.'s  1  and  2  N. 
and  R.'s  1  and  2  W.;  thence  east  on  a  random  line,  set- 
ting temporary  quarter  section  and  section  stakes  to  3. 

If  the  random  line  should  overrun,  or  fall  short,  or 
intersect  the  meridian  north  or  south  of  the  true  cor- 
ner, more  than  3.50  chains,  a  material  error  has  been 
committed,  and  the  line^  must  be  retraced. 

If  the  random  line  should  terminate  within  3.50 
chains  of  the  corner,  measure  the  distance  at  which 
the  meridian  is  intersected  north  or  south  of  the  cor- 
ner, calculate  a  course  which  will  run  a  true  line  back 
from  the  corner  to  the  point  from  which  the  random 
line  started,  measure  westward  to  4,  which  is  the  same 
point  as  2,  establish  the  permanent  corners,  obliterate 
the  temporary  corners  on  the  random  line,  and  throw 
the  excess  or  defect,  if  any,  on  the  west  end  of  the  line. 

In  like  manner,  measure  from  4  to  5,  from  5  to  6, 
from  6  to  7,  and  so  on  to  14,  on  the  standard  parallel, 
throwing  the  excess  or  deficiency  on  the  last  half  mile. 
At  the  intersection  with  the  standard,  parallel,  estab- 
lish the  township  closing  corner,  measuring  and  re- 
cording the  distance  to  the  nearest  standard  corner  on 
said  standard  parallel. 

If  from  any  cause  the  standard  parallel  has  not 
been  run,  the  surveyor  will  plant  the  corner. of  the 


PUBLIC  LANDS.  229 

township   in   place,  subject   to   removal  north  or  south 
when  the  standard  parallel  shall  have  been  run. 

The  surveyor  then  proceeds  to  the  S.  W.  corner  of 
T.  1  N.,  R.  2  W.,  on  the  base  line  at  15,  and  proceeds 
in  a  similar  manner  with  another  range  of  townships, 
and  so  on. 

2.  For  townships  east  of  the  meridian,  begin  at  the 
S.  E.  corner  of  T.  1  A7".,  R.  1  £".,  at  1  on  the  base  line, 
and  proceed  as  on  the  west  of  the  meridian,  except 
that  the  random  lines  are  run  west  and  the  true  lines 
east,  throwing  the  excess  over  480  chains,  or  the  de- 
ficiency, on  the  west  end  of  the  line  in  measuring  the 
first  quarter  section  boundary  on  the  north,  the  remain- 
ing distances  will  be  exact  half-miles  and  miles. 

With  the  field  notes  of  the  exterior  or  township 
lines,  a  plot  of  the  lines,  run  on  a  scale  of  2  inches 
to  the  mile,  must  be  submitted,  on  which  are  noted 
all  objects  of  topography,  which  will  illustrate  the 
notes,  as  the  direction  of  streams,  by  arrow-heads 
pointing  down  stream,  the  intersection  of  the  lines 
by  lakes,  streams,  ponds,  marshes,  swamps,  ravines., 
mountains,  etc. 

259.    Subdivision  or  Section  Lines. 

The  deputy  employed  to  run  the  exterior  lines  of  a 
township  is  not  allowed  to  subdivide  it,  but  another 
is  employed  to  do  this  work,  that  the  one  may  be  a 
check  to  the  other,  thus  securing  greater  accuracy. 

Before  subdividing  a  township,  the  surveyor  must 
ascertain  and  note  the  change  in  the  variation  of  the 
needle  which  has  taken  place  since  the  township  lines 
were  run,  and  adjust  his  compass  to  a  variation  which 
will  retrace  the  eastern  boundary  of  the  township. 


230 


SURVEYING. 


He  must  also  compare  his  own   chaining  with  the 

original    by    remeasuring    tne    first    mile    both   of  the 

south   and   east    lines   of   the    township,  and    note  the 
discrepancies,  if  any. 

The  following  is  a  diagram  of  a  township: 


H 

N 

51 

M 

17 

6 

5 

4 

3 

2 

1 

94 

67 

jC 

:J.3 

16 

93                 92 

90            sn 

65               UG 

48                40 

31               32 

14               15 

7 

8 

9 

10 

II 

12 

89 

64 

47 

30 

u 

8b            b7 

85               86 

62               63 

45               46 

28                 29 

11                  12 

18 

17 

16 

15 

14 

13 

84 

61 

44 

27 

10 

si                82 

80               til 

59               60 

42                43 

25               26 

8                  9 

19 

20 

21 

22 

23 

24 

79 

58 

41 

24 

7 

78               77 

75               76 

56                57 

39               40 

22                 23 

5                   6 

30 

29 

28 

27 

26 

25 

74 

55 

38 

21 

4 

73               72 

70               71 

53               54 

36               37 

19                 20 

2                     3 

31 

32 

33 

34 

35 

36 

69 

52 

35 

18 

i 

The  sections  are  designated  by  beginning  at  the 
N.  E.  corner  and  numbering  west,  1,  2,  3,  4,  5,  6,  then 
east  on  the  next  tier,  7,  8,  . . . ,  then  west,  and  so  on. 

In  running  the  subdivision  lines,  begin  on  the  south    i 
line  of  the   township,  at   the   first  section   corner  west 
of  the  east  line,  numbered  1  in  the  diagram,  and  com- 
mon to  sections  35  and  36. 

Measure  thence  due  north  40  chains,  at  which  point 
establish  a  quarter  section  corner;  thence  due  north 
another  40  chains  to  2,  where  establish  a  section  cor- 
ner common  to  sections  25,  26,  35,  and  3§. 


PUBLIC  LANDS.  231 

Run  a  random  line  from  2  due  east  to  the  township 
line,  setting  up  a  temporary  quarter  section  stake  40 
chains  from  2. 

If  the  random  line  intersect  the  township  line  pre- 
cisely at  the  pre-established  section  corner  at  3,  it  may 
be  established  as  the  true  line  by  blazing  back  and 
making  the  quarter  section  corner  permanent. 

If  the  random  line  intersect  the  township  line  either 
north  or  south  of  the  section  corner,  measure  and  note 
the  distance  of  the  intersection  from  said  corner,  and 
calculate  a  course  which  will  run  a  true  line  from  the 
corner  back  to  4,  where  the  random  line  started. 

Let  A   correspond  to  sec- 
tion  corner   2,  B  to   3,  and  n 
C  to  the  intersection  of  the                                               B 
township  and  random  lines, 
and  north,  for  example,  of  B  the  section  corner. 

BC 

Then,  tan  A  =  --r=-  • 
A.JJ 

Let  /  —  the  number  of  links  in  5(7,  and  m  the  num- 
ber of  minutes  in  A.  Then,  practically,  we  shall  have, 

If  AB  =  I  mile,  m=  I  —  }  I 

If  AB  =  1  mile,  m  =  ±l  —  -&  I. 

If  AB  ^  3  miles,  m  =  JJ. 

If  AB  =  6  miles,  m  =  \  of  \  I 

Let  us  suppose  that  we  have  found  A  --  10J'. 

Now,  as  CA  is  west  by  the  compass,  BA  is  N.  89° 
49V  W.  Run  this  line  and  establish  the  quarter  sec- 
tion at  a  point  equidistant  from  the  two  section  cor- 
ners, which  will  be,  with  sufficient  accuracy,  one-half 
the^  length  of  the  random  line  from  2.  Pull  up  the 
temporary  quarter  section  stake  on  the  random  line. 


232  SURVEYING. 

Proceed  from  4  to  5,  then  on  a  random  line  to  6,  and 
back  on  a  true  line  to  7,  and  so  on  to  16. 

From  16- run  due  north  on  a  random  line  to  the  north 
line  of  the  township,  setting  up  a  temporary  quarter 
section  stake  at  40  chains. 

If  the  random  line  intersect  the  north  line  of  the 
township  at  the  pre-established  section  corner,  the'  ran- 
dom line  will  be  the  true  line,  and  is  made  permanent 
by  blazing  back,  and  making  the  quarter  section  cor- 
ner permanent. 

If  the  random  line  does  not  close  exactly  on  the 
pre-established  section  corner,  measure  and  note  the 
distance  of  the  intersection  from  said  corner,  calculate 
a  course  that  will  run  a  true  line  southward  from  the 
corner  to  16,  run  this  line,  and  establish  the  quarter 
section  corner  on  it  just  40  chains  from  16,  throwing 
the  excess  or  deficiency,  if  any,  on  the  last  half  mile. 

If  the  north  township  line  is  a  base  line  or  stand- 
ard parallel,  no  random  line  is  run,  but  a  true  line  due 
north,  on  which  a  quarter  section  post  is  established 
40  chains  from  16;  and  at  the  intersection  with  said 
base  line  or  standard  parallel,  establish  a  closing  cor- 
ner, measuring  and  noting  its  distance  from  the  corre- 
sponding standard  corner. 

Pass  from  17  to  18,  and  survey  the  second  tier  of 
sections  in  the  same  manner  as  the  first,  closing  on 
the  interior  section  corners  before  established  as  upon 
those  on  the  east  line  of  the  township. 

In  running  the  line  between  the  fifth  and  sixth 
tiers  of  sections,  not  only  is  a  random  line  run  east 
as  before,  but  one  is  run  west  to  the  range  line,  and  a 
true  line  run  back,  and  the  permanent  quarter  section 
corner  established  on  it  just  40  chains  from  the  in- 


PUBLIC  LANDS.  233 

terior  corner,  throwing  the  excess  or  deficiency  on  the 
west  half  mile. 

The  Surveyor-General  furnishes  the  outline  of  the 
diagram,  and  the  deputy  fills  it  out,  and  makes  the 
appropriate  topographical  sketches. 


260.   Meandering. 

Navigable  rivers,  lakes,  and  bayous,  being  public 
highways,  are  meandered  and  separated  from  the  ad- 
joining land. 

Standing  with  the  face  down  stream,  the  bank  on 
the  right  hand  is  called  the  right  bank;  the  bank  on 
the  left,  the  left  bank. 

.  If  a  river  is  navigable,  both  banks  are  meandered, 
care  being  taken  not  to  mistake,  in  high  water,  the 
border  of  bottom-land  for  the  true  bank. 

Commence  at  a  meander  corner  of  the  township 
line,  take  the  bearing  along  the  bank  of  the  river, 
and  measure  the  distance  of  the  longest  possible 
straight  course  to  the  nearest  chain,  if  the  distance 
exceeds  10  chains;  otherwise,  to  the  nearest  ten  links; 
and  so  on  to  the  next  meander  corner  on  another 
boundary  line  of  the  township. 

Enter  in  the  field  book,  after  the  township  notes,  keep- 
ing the  notes  separate  through  each  fractional  section, 
the  date,  the  point  of  beginning,  the  bearings  and  dis- 
tances in  order,  the  intersections  with  all  intermediate 
meander  corners,  the  height  of  falls,  the  length  of  rapids, 
the  location  and  width  at  the  mouth  of  streams  run- 
ning into  the  water  you  are  meandering,  the  location 
of  springs  on  the  banks,  the  nature  of  their  waters, 
the  location  of  islands,  the  elevation  of  banks,  etc. 
S.  N.  20. 


234  SUE  VEYING. 

If  the  river  is  not  navigable,  meander  the  right 
bank,  unless  it  presents  formidable  obstacles  not  found 
on  the  left,  bank;  but  the  crossing  of  the  stream,  in 
meandering,  must  be  made  from  a  pre-established  me- 
ander corner  on  one  bank  to  the  corner  on  the  other 
bank,  and  the  width  of  the  river  between  the  corners 
computed  trigonometrically. 

Wide  flats,  whose  area  is  more  than  40  acres,  per- 
manently covered  with  water,  along  rivers  not  navi- 
gable, are  meandered  on  both  banks. 

The  position  of  islands  in  rivers  is  determined  by 
measuring,  on  or  near  the  bank,  a  base  line,  connected 
with  the  surveyed  lines,  and  taking  the  proper  bear- 
ings to  a  flag  or  other  object  on  the  island,  and  comput- 
ing the  distance  from  the  meander  corners  of  the  river 
to  points  on  the  bank  of  the  island.  The  island  can  be 
meandered  from  such  points. 

In  meandering  lakes,  ponds,  or  bayous,  commence 
at  a  meander  corner  of  the  township  line,  and  proceed 
as  in  case  of  a  river.  If,  however,  the  body  of  water 
is  entirely  within  a  township,  begin  at  a  meander  cor- 
ner established  in  subdividing. 

In  meandering  a  pond  lying  entirely  within  the 
boundaries  of  a  section,  run  to  the  pond  two  lines 
from  the  nearest  section  or  quarter  section  corners,  on 
opposite  sides  of  the  pond,  giving  their  bearings  and 
distances,  and  at  the  intersection  of  these  lines  with 
the  bank  of  the  pond  establish  witness  points  by 
planting  posts,  witnessed  by  bearing  trees  or  mounds 
and  pits,  then  commence  to  meander  at  one  of  these 
points,  and  proceed  around  to  the  other,  and  thence  to 
the  point  of  beginning. 

No  blazes  or  marks  are  made  on  meander  lines  be- 
tween established  corners. 


PUBLIC  LANDS'.  235 

261.    Swamp  Lands. 

By  the  act  of  Congress  approved  Sept.  28th,  1850, 
swamp  and  overflowed  lands,  unfit  for  cultivation,  are 
granted  to  the  state  in  which  they  are  situated. 

If  the  larger  part  of  the  smallest  legal  subdivision 
is  swamp,  it  goes  to  the  state;  if  not,  it  is  retained 
by  the  Government. 

In  order  to  determine  what  lands  fall  to  the  state 
under  the  swamp  act,  it  is  required  that  the  field 
notes,  beside  other  things  required  to  be  noted,  should 
indicate  the  points  where  the  public  lines  enter  and 
leave  all  such  land. 

The  aforesaid  grant  does  not  embrace  lands  subject 
to  casual  inundation,  but  those  only  .where  the  over- 
flow would  prevent  the  raising  of  crops  without  arti- 
ficial aid,  such  as  levees,  etc.  The  surveyor  should 
therefore  state  whether  such  lands  are  continually  and 
permanently  wet,  or  subject  to  overflow  so  frequently 
as  to  render  them  totally  unfit  for  cultivation. 

The  depth  of  inundation  is  to  be  stated,  as  deter- 
mined from  indications  on  the  trees,  and  the  frequency 
of  inundation  should  be  given  as  accurately  as  pos- 
sible, from  the  nature  of  the  case  or  reliable  testimony. 

The  character  of  the  timber,  shrubs,  plants,  etc., 
growing  on  such  lands,  and  on  the  land  near  rivers, 
lakes,  or  other  bodies  of  water,  should  bo  stated. 

The  words  "unfit  for  cultivation"  should  be  em- 
ployed, in  connection  with  the  usual  phraseology,  in 
the  notes,  on  entering  or  leaving  such  lands. 

If  the  margin  of  bottoms,  swamps,  or  marshes,  in 
which  such  uncultivable  land  exists,  is  not  identical 
with  the  body  of  land  unfit  for  cultivation,  a  separate 
entry  must  be  made  opposite  the  marginal  distance. 


236  SURVEYING. 

In  case  the  land  is  overflowed  by  artificial  means, 
such  as  dams  for  milling,  logging,  etc.,  such  overflow 
will  not  be  officially  regarded,  but  the  lines  of  the 
public  surveys  will  be  continued  across  the  same  with- 
out setting  meander  posts,  stating  particularly  in  the 
notes  the  depth  of  the  water,  and  how  the  overflow 
was  caused. 

2G2.    Field  Books. 

The  field  books  are  the  original  and  official  records 
of  the  location  and  boundaries  of  the  public  lands, 
and  afford  the  elements  from  which  the  plots  are 
constructed. 

They  should,  therefore,  contain  an  accurate  record  of 
every  thing  officially  done  by  the  surveyor,  pursuant  to 
instructions  in  running,  measuring,  and  marking  lines, 
and  establishing  corners,  and  should  present  a  full 
topographical  description  of  the  tract  surveyed. 

There  are  four  distinct  field  books. 

1.  A  field   book   for   the   meridian  and  base  lines,  ex- 
hibiting  the   establishment    of   the    township,   section, 
and  quarter  section  corners  on  these  lines,  the  crossing 
of  streams,  ravines,  hills,  and  mountains,  the  character 
of  the  soil,  timber,  minerals,  etc. 

2.  A  field  book  for  standard  parallels  or  correction  lines, 
showing  the  township,  section,  and  quarter  section  cor- 
ners on  the  lines,  and  the  topography  of  the   country 
through  which  the  lines  pass. 

3.  A  field  book   for  exterior  or  township  lines,  showing 
the  establishment  of  corners  on  the  lines,  and  the  to- 
pography. 

4.  A  field  book   for  subdivision  or  section  lines,  giving 
the  corners  and  topography  as  aforesaid. 


PUBLIC  LANDS.  237 

The  variations  of  the  needle  must  be  stated  in  a 
separate  line,  preceding  the  notes  of  measurement, 
which  must  be  recorded  in  the  order  in  which  the 
work  is  done,  and  the  date  must  immediately  follow 
the  notes  of  each  day's  work. 

The  exhibition  of  every  mile  surveyed  must  be  com- 
plete in  itself,  and  be  separated  from  the  preceding  and 
following  notes  by  a  line  drawn  across  the  paper. 

The  topographical  description  must  follow  the  notes 
for  each  mile,  and  not  be  mixed  up  with  them. 

No  abbreviations  are  allowed,  except  for  words  con- 
stantly occurring,  as  sec.  for  section,  ch.  for  chains,  ft. 
for  feet,  J-  sec.  cor.  for  quarter  section  corner. 

Proper  names  are  never  to  be  abbreviated. 

The  field  books  must  be  so  kept  as  to  show  the 
amount  of  work  done  in  each  fiscal  year. 

The  notes  should  be  expressed  in  clear  and  precise 
language,  and  the  writing  legible. 

No  record  is  to  be  obliterated,  or  leaf  mutilated  or 
taken  out. 

The  title-page  of  each  book  should  designate  the 
kind  of  lines  run,  giving  prominently  the  name  of 
the  state  or  territory  and  surveyor,  the  dates  of  con- 
tract, and  of  commencing  and  completing  the  work. 

The  second  page  should  contain  the  names  and 
duties  of  assistants;  and  whenever  a  new  assistant  is 
employed,  or  the  duties  of  any  of  them  changed,  such 
facts,  with  the  reason,  should  be  stated  in  an  appro- 
priate entry,  immediately  preceding  the  notes  taken 
under  such  changed  arrangements. 

An  index,  in  the  form  of  a  diagram  or  plot  of  the 
survey,  with  number  on  each  line,  referring  to  the 
page  of  the  field  notes  on  which  -is  found  the  descrip- 
tion of  the  line,  must  accompany  the  notes. 


238  SURVEYING. 

263.    Records  in  the  Field  Book. 

1.  General  heading  of  the  pages. — The  number  of  the 
township  and   range,   and   the   name   of  the   principal 
meridian  of  reference,  stand  at  the  head  of  each  page. 

2.  Heading  for  each  mile. — The  bearing,  location,  and 
kind   of   line   run,  whether    random   or   true,  must  be 
stated  in  a  line ;   and  the  variation  of  the  needle,  in  a 
separate  line  on  the  page  at  the  head  of  the  notes,  for 
each  mile  run. 

3.  Courses  and  distances.  —  The  course  and  length  of 
each   line   run,   noting   all   necessary  offsets   therefrom, 
with  the  reason  and  mode  thereof. 

4.  The  method  of  perpetuating  corners. — If  a  tree,  note 
the  kind  and  diameter;    if  a  stone,  its  dimensions,  as 
factors  in  the  order  of  length,  breadth,  and  thickness; 
if  a  post,  its  dimensions,  the  kind  of  timber,  the  kind  of 
memorial,  if  any,  buried  by  its  side,  and  .if  surrounded 
by  a  mound,  the  material  of  which  the  mound  is  con- 
structed, whether  of  stones   or   earth ;   the   course   and 
distance   of   the    pits   from   the   center   of   the   mound 
where  a  necessity  exists  for  deviating  from  the  general 
rule  of  witness  trees. 

5.  Bearing  trees. — The  kind  and  diameter  of  all  bear- 
ing trees,  with   the   course   and  distance   of  the  same 
from  their  respective   corners,  and   the  precise  relative 
position    of    the   witness   corners   with    respect    to   the 
true  corners. 

6.  Line  trees. — The   kind,  diameter,  and  distance  on 
the  line,  from   the   corner,  of  all  trees  which  the  line 
intersects. 

7.  Intersection  of  land  objects. — The  distance  at  which 
the  line  first  intersects  and  then  leaves  every  settler's 
claim  and  improvement,   prairie,  bottom-land,  swamp, 


PUBLIC  LANDS.  239 

marsh,  grove,  or  windfall,  with  the  course  of  the  same  at 
both  points  of  intersection;  the  distance  at  which  a  line 
begins  to  ascend,  arrives  at  the  top,  or  reaches  the  foot 
of  all  remarkable  hills  and  ridges,  with  their  courses  and 
estimated  height  above  the  surrounding  country. 

8.  Intersection    of    water    objects. —  The    distance    at 
which  the  line  intersects  rivers,  creeks,  or  other  bodies 
of  water,   the   width   of    navigable    streams,  and   small 
lakes  or  ponds  between  the  meander  corners,  the  height 
of  banks,  the  depth  and  nature  of  the  water. 

9.  Surface. — Level,  rolling,  broken,  or  hilly. 

10.  Soil. — First,  second,  or  third-rate;  clay,  sand,  loam, 
or  gravel. 

11.  Timber.  —  Kind,  in  order  of   abundance,  and  un- 
dergrowth. 

12.  Bottom-lands. — Wet   or   dry;    whether   subject   to 
inundation,  and  to  what  depth. 

13.  Springs.  —  Fresh,   saline,  or   mineral;    and  course 
of  their  streams. 

•  14.  Improvements.— Towns  and  villages,  Indian  vil- 
lages and  wigwams,  houses  and  cabins,  fields,  fences, 
sugar-tree  groves,  mill-seats,  forges  or  factories. 

15.  Coal  beds.  —  Note   the   quality  of  coal   beds,  and 
their  extent  to  the  nearest  legal  subdivision. 

16.  Roads   and  trails. — Whence,  whither,  and   direc- 
tion. 

17.  Rapids,  cascades, —Length   of  rapids,   height  of 
falls  in  feet. 

18.  Precipices.  —  Describe    precipices,   caves,    ravines, 
sink-holes. 

19.  Quarries. — Whether  marble,  granite,  lime-stone  or 
Band-stone. 


240  SURVEYING. 

20.  Natural  curiosities.  —  Interesting   fossils,   ancient 
works,  as  mounds,  fortifications,  embankments,  etc. 

21.  Change  of  variation. — Any  material  change  in  the 
variation  of  the   needle  must  be  noted,  and  the  exact 
points  where  such  variation  occurs. 

22.  Dates.  —  State  the  date  of  each   day's  work   in  a 
separate  line,  immediately  after  the  notes  for  that  day. 

23.  General   description.  —  At    the    conclusion    of   the 
notes  for   the   subdivisional  work,  taken   on    the   line, 
the   deputy  must   subjoin  a  general   description  of  the 
township  in  the  aggregate,  in  reference  to  the  face  of 
the  country,  its  soil,  timber,  geological  features,  etc. 

24.  Verification  of  Deputy  Surveyor.— The  deputy  must 
append  to  each  separate  book  of  field  notes  his  affidavit 
that  all  the  lines  therein  described  have  been  run,  and 
all  the  corners  established  and  perpetuated  according  to 
the  instructions  and  laws,  and  that  the  foregoing  notes 
are  the  true  and  original  field  notes  of  such  survey. 

25.  Verification  of  Assistants.— The  compassman,  flag- 
man, chainmen,  and  axman  must  also  attest,  under  oath, 
that   they  assisted  said  deputy  in   executing  said  sur- 
veys, and  that,  to  the  best  of  their  knowledge  and  be- 
lief, the  work  has  been  strictly  performed  according  to 
the  instructions  furnished  by  the  Surveyor-General. 

26.  Approval  and  certificate  of  the  Surveyor-General.— 

The  Surveyor-General  will  attach  his  official  approval 
to  each  of  the  original  field  books,  and  affix  his  of- 
ficial certificate  to  the  copies  of  the  field  notes  trans- 
mitted to  the  general  land  office,  that  they  are  true 
copies  of  the  originals  on  file  in  his  office. 

The  following  specimen  pages  of  field  notes,  taken 
from  the  United  States  Manual  of  Surveying  Instructions, 
will  illustrate  the  subject : 


PUBLIC  LANDS.  241 


FIELD   NOTES 

OF    THE 

Exterior    and    Subdivision    Twines 

OF  TOWNSHIP  25  NORTH,  RANGE  2  WEST, 
WILLAMETTE  MERIDIAN, 

OREGON. 

Surveyed  by  Robert  Acres,  Deputy  Surveyor, 

Under  his  contract,  dated ?  18 — . 

Survey  commenced . 

Survey  completed    . 


S.  TS.  2L 


242 


SURVEYING. 


264.   Index. 

Referring  the  lines  to  the  pages  of  the  field  notes. 
T.  25  N.,  R.  2  W.,  Willamette  Meridian. 


r 


J 


The   lines   numbered  are   described   in  the   notes  on 
the  pages  indicated  by  the  numbers. 

NAMES  OF  SURVEYOR  AND  ASSISTANTS. 

Robert  Acres,  Surveyor.         George  Sharp,  Axman. 

Peter  Long,  Chainman.         Adam  Dull,  Axman. 

John  Short,  Chainman         Henry  Flagg,  Compassman. 


PUBLIC  LANDS. 


243 


265.  Field  Notes. 


South  Boundary,  T.  25  N.,  R.  2  W.,  Willamette  Meridian. 


Chains. 


Begin  at  the  post,  the  established  corner 
to  Townships  24  and  25  North,  in  Ranges  2 
and  3  West.  The  witness  trees  all  standing, 
and  agree  with  the  description  furnished  me 
by  the  office,  viz : 

A  Black  Oak,  20  in.  dia.,  N.  37°  E.  27  links, 

A  Burr-oak,  24  in.  dia.,  N.  43°  W.  35  links, 

A  Maple,  18  in.  dia.,  S.  27°  W.  39  links, 

A  White  Oak,  15  in.  dia.,  S.  47°  E.  41  links. 

East  on  a  random  line  on  the  South  Bound- 
aries of  sections  31,  32,  33,  34,  35,  and  36. 

Variation  by  Burt's  improved  solar  com- 
pass, 18°  41'  E. 

I  set  temporary  half-mile  and  mile  posts  at 
every  40  and  80  chains,  and  at  5  miles,  74 
chains  53  links,  to  a  point  2  chains  and  20 
links  north  of  the  corner  to  Townships  24 
and  25  North,  Ranges  1  and  2  W. 

(Therefore,  the  correction  will  be  5  chains, 
47  links  West,  and  37  links  South  per  mile.) 

I  find  the  corner  post  standing  and  the 
witness  trees  to  agree  with  the  description 
furnished  me  by  the  Surveyor-General's 
office,  viz: 

A  Burr-oak,  17  in.  dia.,  bears  N.  44°  E.  31 
links, 

A  White  Oak,  16  in.  dia.,  bears  N.  26°  W. 
21  links, 

A  Linden,  20  in.  dia.,  bears  S.  42°  W.  15  Iks., 

A  Black  Oak,-:24  in.  dia.,  bears  S.  27°  E.  14 
links. 


244 


SURVEYING. 


(2) 
South  Boundary,  T.  25  N.,  R.  2  W.,  Willamette  Meridian. 

Chains.  From  the  corner  to  Townships  24  and  25 
X.,  Ranges  1  and  2  W.,  I  run  (at  a  variation 
of  18°  41'  East)  '  [See  Arts.  258,  289.] 

N.   89°  44'  W.,   on   a  true   line  along   the 
40.00       South  Boundary  of  section  36,  set  a  post  for 
quarter  section  corner,  from  which 

A  Beech,  24  in.  dia.,  bears  N.  11°    E.  38 
links  dist. 

A  Beech,  9  in.  dia.,  bears  S.  9°  E.  17  links 
dist. 

62.50  A  Brook,  6  links  wide,  runs  North. 

80.00  Set  a  post  for  corner  to  sections  35  and  36, 

1  and  2,  from  which 

A  Beech,  9  in.  dia.,  bears  N.  22°  E.  16  links 
dist. 

A   Beech,  8   in.  dia.,  bears   N.  19°    W.  14 
links  dist. 

A  White  Oak,  10  in.  dia.,  bears  S.  52°  W. 

7  links  dist. 

A  Black  Oak,  14  in.  dia.,  bears  S.  46°    E. 

8  links  dist. 

Land — level,  good  soil,  fit  for  cultivation. 
Timber — Beech,  various  kinds  of  Oak,  Ash, 
Hickory. 


40.00 


N.  89°  44'  W.,  on  a  true  line  along  the  South 
Boundary  of  section  35,  Variation  18°  41'  E. 

Set  a  post  for  quarter  section  corner,  from 
which 

A  Beech,  8  in.  dia.,  bears  N.  20°  E.  8  links 
dist. 

No  other  tree  convenient;  made  a  trench 
around  post. 


PUBLIC  LANDS.  245 

(3) 
South  Boundary,   T.  25  N.,  R.  2  W.,  Willamette  Meridian. 


Chains. 
65.00 

80.00 


Begin  to  ascend  a  moderate  hill;  bears  N. 
and  S. 

Set  a  post  with  trench,  for  corner  of  sections 
34  and  35,  2  and  3,  from  which 

A  Beech,  10  in.  dia.,  bears  N.  56°  W.  9 
links  dist. 

A  Beech,  10  in.  dia.,  bears  S.  51°  E.  13 
links  dist. 

No  other  tree  convenient  to  mark. 

Land  —  level,  or  gently  rolling,  and  good 
for  farming. 

Timber  —  Beech,  Oak,  Ash,  and  Hickory; 
som?  Walnut  and  Poplar. 


40.00 


80.00 


N.  89°  44'  W.  on  a  true  line  along  the  South 
Boundary  of  section  34,  Variation  18°  41'  E. 

Set  a  quarter  section  post  with  trench, 
from  which 

A  Black  Oak,  10  in.  dia.,  bears  N.  2°  E.  635 
links  dist. 

No  other  tree  convenient  to  mark. 

To  point  for  corner  sections  33,  34,  3  and  4. 

Drove  charred  stakes,  raised  mounds  with 
trenches,  as  per  instructions,  from  which 

A  Burr-oak,  16  in.  dia.,  bears  N.  31°  E.  344 
links. 

A  Hickory,  12  in.  dia.,  bears  S.  43°  W.  231 
links. 

No  other  tree  convenient  to  mark. 

Land  —  level,  rich,  and  good  for  farming. 

Timber  —  some  scattering  Oak  and  Walnut. 


246  SURVEYING. 

(4) 
Smith  Boundary,  T.  25  N.,  R.  2  W.,  Willamette  Township. 


Chains. 

37.51 
40.00 


62.00 
80.00 


N.  89°  44'  W.  on  a  true  line  along  the  South 
Boundary  of  section  33,  Variation  18°  41'  E. 

A  Black  Oak,  24  in.  dia. 

Set  a  post  for  quarter  section  corner,  from 
which 

A  Black  Oak,  18  in.  dia.,  bears  N.  25°  E. 
32  links  dist. 

A  White  Oak,  15  in.  dia.,  bears  N.  43°  W. 
22  links  dist. 

To  foot  of  steep  hill,  bears  N.  E.  and  S.W. 

Set  a  post  for  .corner  to  sections  32,  33,  4 
and  5,  from  which 

A  White  Oak,  15  in.  dia.,  bears  N.  23°  E. 
27  links  dist. 

A  Black  Oak,  20  in.  dia.,  bears  N.  82°  W. 
75  links  dist. 

A  Burr-oak,  20  in.  dia.,  bears  S.  37°  W. 
92  links  dist. 

A  White  Oak,  24  in.  dia,,  bears  S.  26°  E. 
42  links  dist. 

Land  —  gently  rolling ;   rich  farming  land. 

Timber — Oak,  Hickory,  and  Ash. 


37.50 
40.00 


N.  89°  44'  W.  on  a  true  line  along  the  South 
Boundary  of  section  32,  Variation  18°  41'  E. 

A  Creek,  20  links  wide,  runs  North. 

Set  a  granite  stone,  14  in.  long,  10  in.  wide, 
and  4  in.  thick,  for  quarter  section  corner, 
from  which 

A  Maple,  20  in.  dia.,  bears  N.  41°  E.  25 
links  dist. 

A  Birch,  24  in.  dia.,  bears  N.  35°  W.  22 
links  dist. 


PUBLIC  LANDS. 


247 


South  Boundary,  T.  25  JV.,  R.  2  W.,  Willamette  Meridian. 


Chains.  ! 
76.00 


80.00 


To  S.  E.  edge  of  swamp. 

As  it  is  impossible  to  establish  permanently 
the  corner  to  sections  31,  32,  5  and  6,  in  the 
swamp,  I  therefore,  at  this  point,  4.00  chains 
east  of  the  true  point  for  said  section  corner, 
raise  a  witness  mound  with  trench,  as  per 
instructions,  from  which  • 

A  Black  Oak,  20  in.  dia.,  bears  N.  51°  E. 
.115  links.  * 

A  point  in  deep  swamp  for  corner  to  sec- 
tions 31,  32,  5  and*  6. 

Land — rich  bottom;  west  of  creek,  part  wet; 
east  of  creek,  good  for  farming. 

Timber  —  good;  Oak,  Hickory,  and  Walnut. 


11.00 
40.00 


54.00 
57.50 


61.00 
70.00 


N.  89°  44'  W.  on  a  true  line  along  the  South 
Boundary  of  section  31,  Variation  18°  41'  E. 

Leave  swamp  and  rise  bluff  30  feet  high, 
bears  N.  and  S. 

Set  post  for  quarter  section  corner,  from 
which 

A  Sugar  tree,  27  in.  dia.,  bears  S.  81°  W. 
42  links  dist. 

A  Beech,  24  in.  dia.,  bears  S.  71°  E.  24 
links  dist. 

Foot  of  rocky  bluff  30  feet  high,  bears  N.  E. 
and  S.  W. 

A  spring  branch  comes  out  at  the'  foot 
of  the  bluff,  5  links  wide ;  runs  N.  W.  into 
swamp. 

Enter  swamp;    bears  N.  and  S. 

Leave  swamp ;   bears  N.  and  S. 


248 


SURVEYING. 


(6) 
South   Boundary,  T.  25  N.,  R.  2  IF.,  Willamette  Meridian. 


Chains.         The   swamp   contains   about    15   acres,  the 

|   greater  part  in  section  31. 

74.73    I       The    corner   to   Townships  24   and   25   N., 
Ranges  2  and  3  W. 

Land  —  except    the    swamp,   rolling,    good, 
rich  soil. 

Timber  —  Sugar-tree,  Beech,  Swamp  Maple. 
Jan.  25th,  1854. 


8.56 


Between  Ranges  2  and  3  West,  from  corner 
to  Townships  24  and  25  N.,  I  run 

North,  on  the  range  line  between  sections 
31  and  36,  Variation  18°  56'  East. 

Set  a  post  on  the  left  bank  of  Chickeeles 
river,  for  corner  to  fractional  sections  31  and 
36,  from  which 

A  Hackberry,  11  in.  dia.,  bears  N.  50°  E. 
11  links  dist. 

A  Sycamore,  60  in.  dia.,  bears  S.  15°  W.  24 
links-dist. 

I  now  cause  a  flag  to  be  set  on  the  right 
bank  of  the  river,  and  in  the  line  between 
sections  31  and  36.  I  now  cross  the  river, 
and  from  a  point  on  the  right  bank  thereof, 
west  of  the  corner  just  established  on  the 
left  bank,  I  run  North  on  an  offset  line,  25 
chains  and  94  links,  to  a  point  8  chains  and 
56  links  west  of  the  flag.  I  now  set  a  post 
in  the  place  of  the  flag,  for '  corner  to  frac- 
tional sections  31  and  36,  from  which 

A  Beech,  10  in.  dia.,  bears  X.  2°  E.  12 
links  dist. 


PUBLIC  LANDS.  249 

(7) 
Between  Ranges  2  and  3  W.,  T.  25  N.,  Willamette  Meridian. 


Chains. 

34.50 

40.00 


43.41 
80.00 


A  Black  Oak,  12  in.  dia.,  bears  N.  80°  W. 
16  links  dist. 

The  corner  above  described. 

Set  a  post  for  J  section  corner,  from  which 

A  Burr-oak,  20  in.  dia.,  bears  N.  37°  E.  26 
links  dist. 

A  Black  Oak,  24  in.  dia.,  bears  N.  80°  W. 
16  links  dist. 

A  Black  Walnut,  30  in.  dia. 

Set  a  post  for  corner  to  sections  30,  31,  25, 
and  36,  from  which 

A  Beech,  14  in.  dia.,  bears  N.  20°  E.  14 
links  dist. 

A  Hickory,  9  in.  dia.,  bears  N.  25°  W.  12 
links  dist. 

A  Beech,  16  in.  dia.,  bears  S.  40°  W.  16 
links  dist. 

A  White  Oak,  10  in.  dia.,  bears  S.  44°  E. 
20  links  dist. 

Land — level;  rich  bottom;  not  inundated. 

Timber  —  Oak,  Hickory,  Beech,  and  Ash. 


In  like  manner  all  the  other  Township  lines  are  run. 

General  Description. 

This  township  contains  a  large  amount  of  first-rate 
land  for  farming.  It  is  well  timbered  with  Oak,  Hick- 
ory, Sugar-tree,  Walnut,  Beech,  and  Ash. 

Chickeeles  river  is  navigable  for  small  boats  in  low 
water,  and  does  not  often  overflow  its  banks,  which  are 
from  ten  to  fifteen  feet  high. 

The  township  will  admit  of  a  large  settlement,  and 
should  therefore  be  subdivided,  «„ 


250  SURVEYING. 

(8) 

Field   Notes  of  the   Subdivision  Lines  and  Meanders 

of  Chickeeles  River,  in  Towmhip  25  N., 

R.  2  W.,  Willamette  Meridian. 


Chains. 


40.05 
80.09 


9.19 
29.97 
40.00 


51.00 
76.00 


To  determine  the  proper  adjustment  of 
my  compass  for  subdividing  this  township, 
I  commence  at  the  corner  to  Townships  24 
and  25  N.,  R.  1  and  2  W.,  and  run 

North,  on  a  blank  line  along  the  East 
Boundary  of  section  36,  Variation  17°  51' 
East, 

To  a  point  5  links  west  of  the  quarter 
section  corner. 

To  a  point  12  links  west  of  the  cottier  to 
sections  25  and  36. 

To  retrace  this  line,  or  run  parallel  thereto, 
my  compass  must  be  adjusted  to  a  variation 
of  17°  46'  East. 

Subdivision  commenced  Feb.  1,  1854. 

From  the  corner  to  sections  1,  2,  35,  and 
36,  on  the  South  Boundary  of  the  Township, 
I  run 

North,  between  sections  35  and  36,  Varia- 
tion 17°  46'  East, 

A  Beech,  30  in.  dia. 

A  Beech,  30  in.  .dia. 

Set  a  post  for  quarter  section  corner,  from 
which 

A  Beech,  8  in.  dia.,  bears  N.  23°  W.  45 
links  dist. 

A  Beech,  15  in.  dia,,  bears  S.  48°  E.  12 
links  dist. 

A  Beech,  18  in.  dia. 

A  Sugar-tree,  30  in.  dia. 


PUBLIC  LANDS. 

(9) 
Township  25  JV.,  Range  2  TF.,  Willamette  Meridian. 


251 


Chains. 
80.00 


Set  a  post  for  corner  to  sections  25,  26,  35, 
and  36,  from  which 

A  Beech,   28   in.   dia.,  bears   N.  60°    E.  45 
links  dist. 

A  Beech,  24   in.  dia.,  bears   N.  62°   W.  17 
links  dist. 

A  Poplar,  20  in.  dia.,  bears   S.  70°    W.  50 
links  dist. 

•    A   Poplar,   36   in.  dia.,  bears  S.  66°    E.  34 
links  dist. 

Land  —  level,  second-rate. 

Timber  —  Poplar,    Beech,    Sugar-tree,    and 
some  Oak;  undergrowth — same,  and  Hazel. 


9.00 
15.00 
40.00 

55.00 
72.00 
80.00 


40.00 


East,  on  a  random  line  between  sections 
25  and  36,  Variation  17°  46'  East. 

A  Brook,  20  links  wide,  runs  north. 

To  foot  of  hills,  bears  N.  and  S. 

Set  a  post  for  temporary  quarter  section 
corner. 

To  opposite  foot  of  hill,  bears  N.  and  S. 

A  brook,  15  links  wide,  runs  N. 

Intersected  East  Boundary  at  post  corner  to 
sections  25  and  36,  from  which  corner  I  run 

West,  on  a  true  line  between  sections  25 
and  36,  Variation  17°  46'  East. 

Set  a  post  on  top  of  hill,  bears  N.  and  S., 
from  which 

A  Hickory,  14  in.  dia.,  bears  N.  60°  E.  27 
links  dist. 

A  Beech,  15  in.  dia.,  bears  S.  74°  W.  9 
links  dist. 


252 


SURVEYING. 


(10) 
Township  25  Ar.,  Range  2  IF.,  Willamette  Meridian. 


Chains. 
80.00 


The  corner  to  sections  25,  26,  35,  and  36. 

Land  —  east  and  west  parts,  level,  first-rate; 
middle  part,  broken,  third-rate. 

Timber  —  Beech,  Oak,  Ash,  etc. ;  under- 
growth—  same,  and  Spice  in  the  bottoms. 


7.00 
17.20 
18.05 
23.44 
40.00 


60.15 
80.00 


North,  between  sections  25  and  26,  Vari- 
ation 17°  46'  East. 

A  Poplar,  40  in.  dia. 

A  Brook,  25  links  wide,  runs  N.  W. 

A  Walnut,  30  in.  dia. 

A  Brook,  25  links  wide,  runs  N.  E. 

Set  a  post  for  J  sec.  corner,  from  which 

A  Burr-oak,  36  in.  dia.,  bears  N.  42°  E.  18 
links  dist. 

A  Beech,  30  in.  dia.,  bears  S.  72°  W.  9 
links  dist. 

A  Beech,  30  in.  dia. 

Set  a  post  for  corner  to  sections  23,  24,  25, 
26,  from  which 

A  White  Oak,  14  in.  dia.,  bears  N.  50°  E. 
40  links. 

A  Sugar-tree,  12  in.  dia.,  bears  N.  14°  W. 

31  links. 

A  White  Oak,  13  in.  dia.,  bears  S.  38°  W. 

32  links. 

A  Sugar-tree,  12  in.  dia.,  bears  S.  42°  E. 
14  links. 

Land  —  level  on  the  line;  high  ridge  of 
hills  through  the  middle  of  section  25,  run- 
ning N.  and  S. 

Timber  —  Beech, Walnut,  Ash,  Maple,  etc. 


PUBLIC  LANDS. 


253 


(11) 
Township  25  N.,  Range  2  W.,  Willamette  Meridian. 

Chains,  j       In    like    manner    other    subdivison    lines 
are  run. 


24.00 


Notes  of  the  Meanders  of  a  Small  Lake  in 
Section  26. 

Begin  at  the  J  sec.  cor.  on  the  line  between 
sections  23  and  26,  run  thence  South 

To  the  margin  of  the  lake,  where  set  a 
post  for  meander  corner,  from  which 

A  Beech,  14  in.  dia.,  bears  N.  45°  E.  10 
links  dist. 

A  Beech,  9  in.  dia.,  bears  N.  15°  W.  14 
links  dist. 

Thence  meander  around  the  lake  as  follows  : 

S.  53°  E.  17.75.  At  75  links,  cross  outlet 
to  lake  10  links  wide,  runs  N.  E. 

S.  3°  E.  13.00. 

S.  30'  W.  8.00. 

S.  65°  W.  12.00  to  a  point  previously  deter- 
mined 20.30  chains  North  of  the  quarter  sec- 
tion corner  on  the  line  between  sections  26 
and  35. 

Set  post  meander  corner,  Maple,  16  in.  dia., 
bears  S.  15°  W.  20  links  dist. 

Ash,  12  in.  dia,,  bears  S.  21°  E.  15  links 
dist. 

(      In   this   vicinity   we 

!   discovered    remarkable 
N.  63°  W.  10.00     !  foggil 

'  N.  13°  W.  21.00 


tention  of    naturalists. 


254 


SURVEYING. 


(12) 
Township  25  N.,  Range  2  TF.,  Willamette  Meridian. 


Chains. 


N.  52°   E.  17.30  to  the  place  of  beginning. 
This  is  a  beautiful  lake,  with  well-defined 
banks  from  6  to  10  feet  high. 
Land  —  first-rate. 


Meanders  of  the  left  bank  of  Chickeeles  River. 

Begin  at  the  corner  to  fractional  sections  4  and  33,  in 
the  North  Boundary  of  the  Township,  and  on  the  left 
and  S.  E.  bank  of  the  river,  and  run  thence  down  the 
stream  with  the  meanders  of  the  left  bank  of  said  river, 
in  fractional  section  4,  as  follows : 

Remarks. 


To  the  corner  to  fractional  sections 
4  and  5 ;  thence  in  section  5, 


Courses. 

Dist. 

S.76°W. 

18.50 

S.61°W. 

10.00 

S.59°W. 

8.30 

S.54°W. 

10.70 

S.40°W. 

5.60 

S.50°W. 

8.50 

S.37°W. 

17.00 

S.44°W. 

22.00 

S.38°W. 

26.72 

S.21°W. 

16.00 

S.10°W. 

13.00 

South 

8.50 

S.9°E. 

5.00 

S.17°E. 

20.00 

S.10°E. 

12.00 

S.22J°E. 

8.46 

To  the  corner  to  fractional  sections 
5  and  8;  thence  in  section  8, 


To  the  head  of  rapids. 


To  the  foot  of  rapids. 
To  the  corner  to  fractional  sections 
8  and  17. 

Land,   along  fractional    section  8, 


PUBLIC  LANDS. 

(13) 
Township  25  N.,  Range  2  W.,  Willamette  Meridian. 


255 


Courses. 

Dist. 

Remarks. 

high,   rich    bottom;    not   inundated. 

The  rapids  are  37.00  chains  long  ; 

rocky  bottom  ;  estimated  fall,  10  feet. 

Meanders  in  Section  17. 

S.17°E. 

15.00 

At  5  chains,  discovered  a  vein  of 

coal,   which    appears    to    be   5    feet 

thick,  and  may  be  readily  worked. 

S.8°E. 

12.00 

S.4°W. 

22.00 

At  3  chains,  the  ferry  across  the 

river  to  Williamsburgh,  on  the  oppo- 

site side  of  the  river. 

S.25°W. 

17.00 

S.78°W. 

12.00 

S.71°W. 

9.55 

To  the  corner  to  fractional  sections 

17  and  18;  thence  in  section  18, 

S.65°W. 

15.00 

S73f°W. 

15.93 

To  the  corner  to  fractional  sections 

18  and  19. 

S.65°W. 

14.00 

In  section  19. 

S.60°W. 

23.00 

S.42°W. 

10.00 

S.20°W. 

10.00 

S16J°W. 

13.83 

JS^r*  At   2  chains,   cross   outlet    to 

pond  and  lake,  50  links  wide,  to  the 

corner  to  fractional  sections  19  and 

24,  on  the  range   line,  32.50  chains 

North  of  the  corner  to  sections   19, 

/ 

30,  24,  and  25. 

The  above  selections  will  serve  as  specimens  of  the 
manner  of  taking  the  field  notes. 


256  SURVEYING. 

266.    General  Description. 

The  quality  of  the  land  in  this  township  is  con- 
siderably above  the  average.  There  is  a  fair  propor- 
tion of  rich  bottom-land,  chiefly  situated  on  both  sides 
of  Chickeeles  river,  which  is  navigable,  through  the 
township,  for  steamboats  of  light  draft,  except  over  the 
rapids  in  Section  8. 

The  uplands  are  generally  rolling,  good  first  and 
second  rate  land,  etc. 

267.   Certificates. 

I,  Robert  Acres,  Deputy  Surveyor,  do  solemnly  swear 
that,  in  pursuance  of  a  contract  with  ,  Surveyor 

of  the  public  lands  of  the  United  States,  in  the  State 
[or  Territory]  of  ,  bearing  date  the  day 

of  ,  18     ,  and  in  strict  conformity  to  the   laws 

of  the  United  States  and  the  instructions  furnished  by 
the  said  Surveyor-General,  I  have  faithfully  surveyed 
the  exterior  boundaries  [or  subdivision  and  meanders, 
as  the  case  may  be]  of  Township  number  twenty-five 
North  of  the  base  line  of  Range  number  two  West  of 
the  Willamette  Meridian,  in  the  aforesaid;  and 

do  further  solemnly  swear  that  the  foregoing  are  the 
true  and  original  field  notes  of  such  survey. 

ROBERT  ACRES, 

Deputy  Surveyor. 

Subscribed  by  said  Robert  Acres,  Deputy  Surveyor, 
and  sworn  to  before  me,  a  Justice  of  the  Peace  for  the 

County,  in  the  State  [or  Territory]  of 
this  day  of  ,  18     . 

HENRY  DOOLITTLE, 

Justice  of  the  Peace. 


PUBLIC  LANDS.  257 

We  hereby  certify  that  we  assisted  Robert  Acres, 
Deputy  Surveyor,  in  surveying  the  exterior  boundaries, 
and  subdividing  Township  number  twenty-five  North 
of  the  base  line  of  Range  number  two  West  of  the 
Willamette  Meridian,  and  that  said  Township  has  been, 
in  all  respects,  to  the  best  of  our  knowledge  and  belief, 
well  and  faithfully  surveyed,  and  the  boundary  monu- 
ments planted  according  to  the  instructions  furnished 
by  the  Surveyor-General. 

PETER  LONG,  Chainman. 

JOHN  SHORT,  Chainman. 

GEORGE  SHARP,  Axman. 

ADAM  DULL,  Axman. 

HENRY  FLAGG,  Compassman. 

Subscribed  and  sworn  to  by  the  above  named  per- 
sons, before  me,  a  Justice  of  the  Peace  for  the  county 
of  ,  in  the  State  [or  Territory]  of  ,  this 

day  of  ,  18     . 

HENRY  DOOLITTLE, 

Justice  of  the  Peace. 

SURVEYOR'S  OFFICE  AT  ,  18    . 

The  foregoing  field  notes  of  the  Survey  of  [here  de- 
scribe the  survey],  executed  by  Robert  Acres,  under  his 
contract  of  the  clay  of  ,  18  ,  in  the 

month  of  ,  18     ,  having  been  critically  examined, 

the  necessary  corrections  and  explanations  made,  the 
said  field  notes,  and  the  surveys  they  describe,  are 
hereby  approved.  A.  B., 

Surveyor-  General. 


To  the  notes  of  each  Township,  in  the  copies  of  the 
field  notes  transmitted  to  the  seat  of  government,  the 
Surveyor-General  will  append  the  following  certificate: 

S.  N.  22. 


258  SURVEYING. 

I  certify  that  the  foregoing  transcript  of  the  field 
notes  of  the  Survey  of  the  [here  describe  the  character 
of  the  surveys,  whether  meridian,  base  line,  standard 
parallel,  exterior  township  lines,  or  subdivision  lines 
and  meanders  of  a  particular  township],  in  the  State 
[or  Territory]  of  ,  has  been  correctly  copied  from 

the  original  notes  on  file  in  this  office.         A.  B., 

Surveyor-  General. 

268.   Corners  and  Boundaries  Unchangeable. 

According  to  an  act  of  Congress,  entitled  "An  act 
concerning  the  mode  of  Surveying  the  Public  Lands 
of  the  United  States,"  approved  February  llth,  1805, 
and  still  in  force, 

1st.  "All  the  corners  marked  in  the  surveys  returned 
by  the  Surveyor-General,  shall  be  established  as  the 
proper  corners  of  sections  or  subdivisions  of  sections 
which  they  were  intended  to  designate;  and  the  cor- 
ners of  half  and  quarter  sections,  not  marked  on  said 
surveys,  shall  be  placed,  as  nearly  as  possible,  equi- 
distant from  those  two  corners  which  stand  on  the 
same  line." 

2d.  "The  boundary  lines  actually  run  and  marked 
in  the  surveys  returned  by  the  Surveyor-General,  shall 
be  established  as  the  proper  boundary  lines  of  the 
sections  or  subdivisions  for  which  they  were  intended; 
and  the  length  of  such  lines,  as  returned  by  the  Sur- 
veyor-General aforesaid,  shall  be  held  and  considered 
as  the  true  length  thereof." 

If  it  is  afterward  found  that  a  post  is  out  of  line, 
or  that  the  line  has  been  unequally  subdivided,  the 
general  government  only  has  the  power  of  correction, 
and  that  only  while  it  holds  the  title  to  the  lands 
affected. 


PUBLIC  LANDS.  259 

Such  boundaries  only  as  "are  established  by  the  Sur- 
veyor-General, or  the  deputy,  in  the  performance  of 
his  official  duties,  and  in  accordance  with  law,  come 
under  the  above  rules. 


269.   Restoring:  Lost  Boundaries. 

Lost  boundaries  must  be  restored  in  conformity  with 
the  laws  under  which  they  were  originally  established. 

At  an  early  day,  three  sets  of  section  corners  were 
established  on  the  range  lines;  later,  two  sets  on  all  the 
township  boundaries;  at  present,  the  section  lines  close 
on  previously  established  corners  on  township  corners, 
making  one  set  of  corners,  except  on  the  base  lines 
and  standard  parallels,  where  double  corners  —  standard 
corners  and  closing  corners  —  are  established. 

In  order  to  restore  lost  boundaries  correctly,  the 
surveyor  must  know  the  manner  in  which  townships 
were  originally  subdivided. 

In  case  of  three  sets  of  corners  on  the  range  lines, 
one  set  was  planted  when  the  exteriors  were  run. 

Corners  on  the  east  and  west  lines  between  two  town- 
ships, belong  to  the  sections  of  the  township  north. 

From  these  corners,  section  lines  were  run  due  north, 
which  would  not,  in  general,  close  on  the  corners  of 
the  township  line  on  the  north,  thus  making  two  sets 
of  corners  on  the  north  and  south  boundaries  of  the 
township. 

The  east  and  west  lines  were  run  due  east  and  west 
from  the  last  interior  section  corner,  and  new  corners 
established  at  the  intersections  with  the  range  lines. 

In  case  of  two  sets  of  corners,  the  subdivisions  were 
made  as  &bove,  except  that  the  east  and  west  lines 


260  SURVEYING. 

were  closed  on  the  corners  previously  established  on 
the  east  boundary,  but  were  run  due  west  from  the 
last  interior  section  corner  to  the  range  line,  and  new 
section  corners  established  at  the  intersection  with  the 
range  line. 

The  method  of  making  but  one  set  of  corners,  ex- 
cept on  the  base  line  and  standard  parallels,  is  the  one 
now  in  vogue,  and  has  been  sufficiently  considered. 

270.   Restoring  Lost  Corners. 

Lost  corners  must  be  restored,  if  possible,  to  their 
exact  original  position. 

The  surveyor  should  seek  to  accomplish  this,  first, 
by  the  aid  of  bearing  trees,  mounds,  etc.,  described  in 
the  original  field  notes. 

If  the  corner  can  not  be  located  in  this  way,  good 
testimony  may  be  taken. 

It  often  happens  that  in  retracing  lines,  the  meas- 
urements do  not  agree  with  the  field  notes.  When 
such  cases  occur,  from  whatever  cause,  the  surveyor 
must  establish  his  corners  at  intervals  proportional  to 
those  given  in  the  original  field  notes. 

1.   To  restore  a  lost  corner  common  to  four  sections. 

Find  the  distances  between  the  nearest  noted  line 
trees  or  well-defined  corners,  north  and  south,  and  east 
and  west  of  the  lost  corner.  Establish  the  corner  be- 
tween them  at  a  point  intercepting  distances  propor- 
tional to  those  given  in  the  original  notes. 

2.   To  restore  one  of  a  double  corner  when  the  other  is  standing. 

First  ascertain  to  which  sections  the  existing  cor- 
ner belongs.  Then  re-establish  the  lost  corner  in  the 


PUBLIC  LANDS.  261 

direction  and  at  the  distance  stated  in  the  original 
notes.  Verify  the  work  by  chaining  to  noted  line 
trees  or  corners,  having  previously  compared  your 
chaining  with  that  of  the  United  States  deputy  by 
rechaining  between  corners  noted  in  the  original  sur- 
vey, and  making  all  distances  proportional. 

3.  To  restore  that  one  of  a  double  corner  established  in  run- 

ning the  township  lines  when  both  are  missing. 

Run  a  straight  line  between  tttg  nearest  noted  line 
trees  or  corners  on  the  line,  and,  at  the  distance  given 
in  the  notes,  establish  the  corner  which  will  be  com- 
mon to  two  sections  north  or  west  of  the  line. 

Let  the  accuracy  of  the  result  be  verified  by  measur- 
ing to  the  next  section  corner  west  or  north. 

4.  To  restore  that  one  of  a  double  corner  established  in  subdi- 

viding the  toivnship  when  both  are  missing. 

Retrace  the  section  line  which  closed  on  the  corner, 
and  establish  the  section  post  at  the  intersection  with 
the  township  line.  Verify  the  result  by  measuring  on 
the  township  line  to  noted  objects. 

The  restored  corner  will  be  common  to  two  sections 
south  or  east  of  the  line. 

5.  To  restore  one  of  a  triple  corner,  on  a  range  line  when 

one  at  least  remains  standing. 

The  one  of  the  triple  corner,  established  when  the 
range  line  was  run,  is  not  a  section  corner. 

First  identify  the  existing  corners,  then  establish 
the  lost  corner,  according  to  the  field  notes,  north  or 
south  of  the  existing  corner,  on  the  line,  and  verify 
the  result. 


262  SURVEYING. 

If  the  field  notes  do  not  give  the  distances  between 
the  triple  corners,  retrace  the  section  line  closing  on 
said  corner. 

6.   To  restore  a  triple  corner  when  alt  are  lost. 

Rechain  the  range  line,  and  retrace  the  section  lines 
closing  on  the  range  line. 

7.   To  restore  lost  quarter  section  corners. 

1st.  Except  on  those  section  lines  which  close  on 
the  north  or  west  boundaries  of  a  township,  quarter 
section  corners  are  equidistant  between  the  two  section 
corners.  Hence,  rechain  the  section  line,  then  chain 
back  one-half  the  distance. 

2d.  On  township  lines,  where  there  may  be  double 
section  corners,  only  one  set  of  quarter  section  corners 
are  actually  marked  in  the  field  —  those  established 
when  the  exteriors  are  run  half-way  between  the 
section  corners  established  at  the  same  time.  These 
are  restored  as  above. 

The  same  will  apply  when  there  are  triple  corners. 

3d.  If  the  section  line  closes  on  the  north  or  west 
boundary  of  a  township,  the  quarter  section  corner 
must  be  established  40  chains  of  the  original  measure- 
ment from  the  last  interior  section  corner. 

8.   To  restore  lost  township  corners. 

1st.  If  the  corner  is  common  to  four  townships,  re- 
trace the  township  and  range  lines,  and  establish  trie 
corner  at  their  intersection. 

2d.  If  the  corner  is  common  only  to  two  townships, 
as  may  be  the  case  on  the  base  line  or  standard  paral- 
lels, retrace  the  base  line  or  standard  parallel  from  the 


PUBLIC  LANDS. 


263 


last  standing  corner,  if  the  lost  corner  is  common  to 
two  townships  north;  but  if  the  lost  corner  is  common 
to  two  townships  south,  retrace  also  the  range  line. 

9.   To  restore  lost  meander  corners. 

Retrace  the  lines  which  close  upon  the  banks  in  the 
direction  they  were  originally  run. 

Fractional  section  lines  closing  on  Indian  boundaries, 
private  grants,  etc.,  should  be  retraced,  and  the  corners 
established  in  the  same  manner. 

Remark. —  If,  in  restoring  a  lost  corner,  the  original 
corner  is  found  by  some  unmistakable  trace,  it  must 
stand,  and  the  resurvey  be  made  to  correspond. 


271.    Subdividing  Sections. 

The  United  States  deputy  runs  only  the  exterior  or 
section  lines,  and  makes  the  section  and  quarter  sec- 
tion corners. 

Lines  joining  the  opposite  quarter  section  corners 
divide  the  section  into  quarter  sections  of  160  acres 
each. 

These  quarter  sections  are  di- 
visible into  half-quarters  of  80 
acres,  and  these  into  quarter- 
quarters  of  40  acres. 

These  are  the  legal  subdivis- 
ions of  a  section,  and  are  exhib- 
ited in  the  annexed  diagram. 

If  private  parties  wish  the  subdivision  lines  traced 
on  the  ground,  they  employ  the  county  surveyor,  or  a 
private  surveyor,  who  must  be  governed  by  the  section 
and  quarter  section  corners  previously  established. 


40  A. 

40  A. 

80  A 

80  A- 

40  A. 

40  A. 

80 

A. 

)  A. 

so 

A. 

264  SURVEYING. 

The  following  rules  will  enable  the  surveyor  to  sub- 
divide a  section  in  accordance  with  the  laws  of  the 
United  States : 

1.  The  original   section  and  quarter  section  earners 
must  stand  where  they  were  established  by  the  govern- 
ment surveyor. 

2.  The   quarter-quarter  corners  must   be   established 
equidistant,  and  on   the  line   between  the  section  and 
quarter  section  corners  of  the  exterior  lines  of  the  sec- 
tion, and  equidistant  and  on  the  line  between  quarter 
section  corners  of  internal  lines  of  the  section. 

3.  All  subdivision  lines  must  run  straight  from  the 
proper   corner   in  one   exterior  line   of  the    section   to 
the  corresponding  corner  in  the  opposite  exterior  line. 

4.  In   fractional    sections,   where    no   opposite    corre- 
sponding corner  has  been   established,  the   subdivision 
line  must  be  run  from  the  given  corner  due  north  and 
south,  or   east   and  west,  to   the    exterior   boundary  of 
said  fractional  section. 

5.  Anomalous  sections  or  sections  larger  than  a  mile, 
sometimes   close   on   a   previously  established    line,   in 
finishing  up  a  public  survey. 

Quarter  section  and  section  corners  are  established 
40  chains  and  80  chains,  respectively,  from  the  previ- 
ously established  corners,  and  posts  are  planted  every 
20  chains  of  the  remaining  distance. 

Anomalous  sections  are  subdivided  by  running 
straight  lines  from  the  corners  on  the  south  line  to 
the  corresponding  corners  on  the  north,  and  east,  and 
west  lines,  the  same  as  in  regular  sections. 


VARIATION  OF  THE  NEEDLE.  265 

VARIATION   OF   THE   NEEDLE. 
272.    Definitions  and  Illustrations. 

The  variation  of  the  needle  is  the  angle  which  the 
magnetic  meridian  makes  with  the  true  meridian. 

The  variation  is  east  or  west,  according  as  the  north 
end  of  the  needle  is  east  or  west  of  the  true  meridian. 

The  variation  is  different  at  different  places,  and  it 
does  not  remain  the  same  at  the  same  place. 

The  line  of  no  variation  is  that  line  traced  through 
those  points  on  the  surface  of  the  earth  where  the 
needle  points  due  north. 

At  all  places  east  of  this  line,  the  variation  is 
west;  and  at  all  places  west  of  this  line,  the  variation 
is  east. 

West  variation  is  designated  by  the  sign  phis,  and 
east  variation  by  the  sign  minus. 

In  the  year  1840,  at  a  point  whose  latitude  is  40° 
53',  and  longitude  80°  13',  being  a  little  8.  E.  of  Cleve- 
land, O.,  the  variation  was  nothing.  The  line  of  no 
variation  passed  through  this  point  N.  24°  35'  W.,  and 
S.  24°  35'  E. 

273.   Changes  of  Variation. 

1.  Irregular   changes. —  The  needle  is  subject  to  sud- 
den changes  coincident,  in  time,  with  a  thunder  storm, 
an  aurora  borealis,  solar  changes,  etc. 

2.  Diurnal   changes,  —  In    the    northern    hemisphere, 
the    north    end   of   the    needle    moves   from    10'  to   15' 
west  from  about  8  A.  M.  to  2  P.  M.,  and  then  gradu- 
ally returns  to  its  former  position. 

S.  N.  23. 


266 


SURVEYING. 


3.  Annual  changes. — The   diurnal  changes  vary  with 
the  season,  being  about  twice   as   great  in  the  summer 
as  in  the  winter. 

4.  Secular  changes. — In  addition  to  the  above  changes, 
there  is  a  change  of  variation,  in  the  same  direction, 
running  with  considerable  regularity  through  a  period 
of  about   234   years,  as  is  indicated  by  observations  at 
Paris. 

In  the  United  States,  the  north  end  of  the  needle 
was  moving  east  from  the  earliest  recorded  observa- 
tions till  about  the  year  1810,  since. which  time  the 
movement  has  been  west,  at  the  rate,  on  an  average, 
of  about  5'  per  annum. 

We  give  the  following  tables  of  places,  their  latitude 
and  longitude,  and  variation  as  it  was  in  1840,  and  the 
annual  change  of  variation,  from  the  tables  prepared 
by  Professor  Loomis  for  the  39th  and  42d  volumes  of 
Silliman's  Journal: 


Places  near  the  Line  of  no   Variation. 


Places. 

Lat. 

Lon. 

Var. 

An.  Mo.  \ 

A  Point. 

40°  53' 

80°  13' 

0°00' 

4-  4'.4 

Cleveland,  O. 

41°  31' 

81°  45' 

-0°19' 

4'.4 

Mackinaw. 

45°  51' 

84°  41' 

—  2°  08' 

3'.9 

Charlottesville,Va. 

39°  02' 

78°  30' 

+  0°  19' 

3'.7 

Assuming  the  annual  motion  uniform,  and  correctly 
found  for  1840,  the  variation  for  any  subsequent  time 
can  be  found  by  multiplying  the  annual  motion  by 
the  number  of  years  since  1840,  and  taking  the 
algebraic  sum  of  the  product  and  the  variation  at 
that  date. 


VARIATION  OF  THE  NEEDLE. 
Places  where  the  Variation  was  West. 


267 


Places. 

Lat. 

Lon. 

Var. 

An.  Mo. 

Point  in  Maine. 

48°0(y 

67°  37' 

+  19°  30' 

+  8'.8 

Waterville,  Me. 

44°  27' 

69°  32' 

12°  36' 

-5'.7 

Montreal. 

45°  31' 

73°  35' 

10°  18' 

5'.7 

Burlington,  Vt. 

44°  27' 

73°  10' 

9°  27' 

5'.3 

Hanover,  N.  H. 

43°  42' 

72°  14' 

9°  20' 

5'.2 

Cambridge,  Mass. 

42°  22' 

71°  08' 

9°  12' 

5'." 

Hartford,  Conn. 

41°  46' 

72°  41' 

6°  58' 

5'. 

Newport,  R.  I.    - 

41°  28' 

71°  21' 

7°  45' 

5'. 

Geneva,  N.  Y. 

42°  52' 

77°  03' 

4°  18' 

4'.1 

West  Point. 

41°  25' 

74°  00' 

6°  52' 

4'. 

New  York  City. 

40°  43' 

71°  01' 

5°  34' 

3'.6 

Philadelphia. 

39°  57' 

75°  11' 

4°  08' 

3'.2 

Buffalo,  N.  Y. 

42°  52' 

79°  06' 

1°37' 

4'.1 

Places  where  the  Variation  was  East. 


Places. 

Lat. 

Lon. 

Var. 

An.  Mo. 

Jacksonville,  111. 

39°  43' 

90°  20' 

-8°  28' 

-f  2'.5 

St.  Louis,  Mo. 

38°  37'  . 

90°  17' 

8°  37' 

2'.3 

Nashville,  Tenn. 

36°  10' 

86°  52' 

6°  42' 

2'. 

Louisiana. 

29°  40' 

94°  00' 

8°  41' 

1'.4 

Mobile,  Ala. 

30°  42' 

88°  16' 

7°  05' 

1'.4 

Tuscaloosa,  Ala. 

33°  12' 

87°  43' 

7°  26' 

1'.6 

Columbus,  Ga. 

32°  28' 

85°  11' 

5°  28' 

2'. 

Milledgeville,  Ga. 

33°  07' 

83°  24' 

5°  07' 

2'.4 

Savannah,  Ga. 

32°  05'     81°12; 

4°  13' 

2'.7 

Tallahassee,  Fa. 

30°  26' 

84°  27' 

5°  03' 

1'.8 

Pensacola,  Fa. 

30°  24' 

87°  23' 

5°  53' 

1'.4 

Logansport,  Ind. 

40°-  45' 

86°  22' 

5°  24' 

2'.7 

Cincinnati,  0. 

39°  06' 

84°  27' 

4°  46' 

3/1 

268  SURVEYING. 

274.   Methods  of  Ascertaining  the  Variation. 

First   establish  a  true  meridian,  which  may  be  done 

1.  By  means  of  Burfs  Solar  Compass. 

2.  By  observation  of  the  North  star,  when  on  the  meridian. 

The  north  star  is  about  1°  22'  from  the  true  pole, 
around  which  it  revolves  in  a  siderial  day,  or  23  h., 
56  in.,  4  s. 

Twice  in  this  period  the  star  will  be  on  the  meridian. 

The  exact  moment  of  its  passage 
can  be  determined  very  nearly,  from 
the  fact  that  it  reaches  the  meridian 
almost  at  the  same  instant  as  Alioth 
in  the  tail  of  the  Great  Bear,  or  the 
first  star  in  the  handle  of  the  Dipper. 

Suspend   a  plumb  line  a  few  feet 
in  front  of  the  telescope,  and    place         ^ 
a  faint  light  near  the  object  glass  of 
the  telescope,  so  that  the  spider  lines 
may  be  seen. 

Just  17  minutes  after  the  plumb  line,  the  North  star, 
and  Alioth  all  fall  on  the  vertical  spider  line,  the  North 
star  is  on  the  meridian. 

The  horizontal  limb  of  the  instrument  is  then  firmly 
clamped,  and  the  telescope  is  turned  down  horizontally. 

A  light,  shining  through  a  small  aperture  in  a  board, 
at  some  distance,  say  ten  rods,  is  moved  by  an  assistant, 
according  to  signals,  till  it  ranges  with  the  intersection 
of  the  spider  lines. 

A  stake  driven  into  the  ground  directly  under  the 
light,  and  another  directly  under  the  telescope,  will 
mark,  on  the  ground,  the  true  meridian. 


VARIATION  OF  THE  NEEDLE.  269 

The  season  of  the  year  may  be  such  that  Alioth 
may  be  above  instead  of  below  the  North  star,  when 
both  are  on  the  meridian  at  night.  With,  the  telescope, 
the  stars  can  be  seen  in  the  day-time. 

3.  By  the  azimuth  of  the  North  star. 

When  the  North  star  is  farthest  from  the  meridian, 
east  or  west,  it  is  said  to  be  at  its  greatest  eastern  or 
western  elongation. 

The  azimuth  of  a  star  is  the  angle  which  a  vertical 
plane,  through  the  star,  makes  with  the  meridian  plane. 

Let  us  now  find  the  azimuth  of  the  North  star  at  its 
greatest  elongation. 

Let  Z  be  the  zenith,  P  the  pole,  S 
the  North  star  at  its  greatest  elong- 
ation, ZP,  ZS,  and  PS  arcs  of  great 
circles.  Then  ZPS  will  be  a  spherical 
triangle,  right-angled  at  £,  and  the 
angle  Z  will  be  the  azimuth,  PS  the 
greatest  elongation,  and  ZP  the  com- 
plement of  latitude;  since  the  elevation  of  the  pole 
above  the  horizon  is  equal  to  the  latitude. 

Now,  from  Napier's  principles,  we  have 

sin  e  =  cos  I  cos  (90°  — Z). 

„       sin  e 

.  • .     sin  Z  = r  - 

cos  I 

Introducing  R  and  applying  logarithms,  we  have 
log  sin  Z  =  10  -f-  log  sin  e  —  log  cos  I, 

Hence,  the  azimuth  is  readily  computed  if  we  know 
the  greatest  elongation  of  the  star  and  the  latitude  of 
the  place. 


270 


SURVEYING. 

Greatest  Elongation  of  Polaris. 


Date. 

Elongation. 

Date. 

Elongation. 

Date. 

Elongation. 

1870 

1°  23'  01". 

1880 

1°  19'  50".4 

1890 

1°  16'  40".7 

1871 

1°22'41".9 

1881 

0  19'  31".4 

1891 

1°  16'  21".8 

1872 

1°  22'  22".9 

1882 

0  19'  12".5 

1892 

1°  16'  03" 

1873 

1°  22'  03".8 

1883 

0  18'  53".5 

1893 

1°  15'44".l 

1874 

1°  21'  44".8 

1884 

0  18'  34".5 

1894 

1°  15'  25".3 

1875 

1°21"25".7 

1885 

1°  18'  15".5 

1895 

1°  15'06".4 

1876 

1°  21'  06".6 

1886 

1°  17'  56".6 

1896 

1°  14'  47".6 

1877 

1°  20'  47".6 

1887 

1°  17'  37".6 

1897 

1°  14'  28".7 

1878 

1°  20'  28".5 

1888 

1°  17'  18".6 

1898 

1°  14'09".9 

1879 

1°  2(X  09".5 

1889 

1°  16'  59".7 

1899 

1°  13'  51" 

The  elongation  in  the  table  is  given  for  the  1st  of 
January  of  each  year;  but  the  elongation  for  any  month 
of  the  year  can  be  readily  found. 

Thus,  let  us  find  the  elongation  for  May  1st,  1873. 

Jan.  1st,  1873,  Elongation  ==  1°  22'  03".8 
Jan.  1st,  1874,  Elongation  =  1°  21'  44".8 

Change  for  12  months  19" 

Change  for    4  months  6.3" 

.  • .     Then,  for  May  1st,  1873,  we  shall  have, 

Elongation  =  1°  22'  03".8  —  6".3  =  1°  21'  57".5. 

1.  Find  the  azimuth  of  the  North  star  at  its  greatest 
elongation,  May  1st,  1873  —  latitude  40°.     Ans.  1°  47'. 

2.  Find  the  azimuth  of  the  North  star  at  its  greatest 
elongation,  July  1st,  1875— latitude  42°.    Ans.  1°  49J'. 

3.  Find  the  azimuth  of  the  North  star  at  its  greatest 
elongation,  Sept.  21st,  1880  — latitude  45°  45'. 

Ans.  1°  54 £'. 


VARIATION  OF   THE  NEEDLE. 


271 


It  will  be  necessary  to  know  the  times  of  the  greatest 
elongation.  These  times  are  given  in  the  following 
tables,  for  the  1st,  llth,  and  21st  of  each  month  of  the 
year  1880,  which  will  answer  the  purpose  for  the  rest 
of  the  century,  since  the  change  of  time  is  very  slow, 
being  only  about  16  minutes  in  50  years. 

Eastern  Elongation. 


Month. 

1st  day. 

llth  day.' 

21st  day. 

April. 

6h.  40m.  A.M. 

6h.  Olm.  A.M. 

5h.  22m.  A.M. 

May. 

4h.  42m.  A.M. 

4h.  03m.  A.M. 

3h.  24m.  A.M. 

June. 

2h.  41m.  A.M. 

2h.  Olm.  A.M. 

Ih.  22m.  A.M. 

July. 

Oh.  43m.  A.M. 

Oh.  00m.  A.M. 

lib.  21m.  P.M. 

August. 

lOh.  38m.  P.M. 

9h.  59m.  P.M. 

9h.  19m.  P.M. 

Sept. 

8h.  36m.  P.M. 

7h.  57m.  P.M. 

7h.  17m.  P.M. 

Western  Elongation. 


Month. 

1st  day. 

llth  day. 

21st  day. 

Oct. 

6h.  31m.  A.M. 

5h.  52m.  A.M. 

5h.  13m.  A.M. 

Nov. 

4h.  30m.  A.M. 

3h.  50m.  A.M. 

3h.llm.  A.M. 

Dec. 

2h.  31m.  A.M. 

Ih.  52m.  A.M. 

Ih.  13m.  A.M. 

Jan. 

Oh.  28m.  A.M. 

llh.44m.P.M. 

lib.  04m.  P.M. 

Feb. 

lOh.  22m.  P.M. 

9h.  42m.  P.M. 

9h.  03m.  P.M. 

March. 

8h.  31m.  P.M. 

7h.  52m.  P.M. 

7h.  13m.  P.M. 

About  half  an  hour  before  the  greatest  eastern  or 
western  elongation,  place  the  transit  in  a  convenient 
position,  and  level  it  carefully. 

Paste  white  paper  on  a  board  about  one  foot  square, 
and  perforate  the  board  through  the  center  with  a  two- 
inch  auger,  and,  on  the  lower  edge,  fix  some  contriv- 
ance for  holding  a  candle. 


272  SURVEYING. 

Let  this  board  be  fixed  to  a  vertical  staff,  so  as  to 
slide  freely  up  and  down,  and  let  it  be  placed  about 
one  foot  in  front  of  the  telescope,  so  that  the  light 
reflected  from  the  paper  will  render  the  spider  lines 
visible. 

Slide  the  board  up  or  down  the  staff  till  the  North 
star  is  visible  through  the  telescope  and  orifice  in  the 
board,  and  bring  the  vertical  spider  line  in  range  with 
the  star. 

As  the  star  approaches  its  greatest  elongation,  move 
the  telescope  by  a  tangent  screw,  so  as  to  keep  the 
vertical  line  in  range  with  the  star.  When  the  star 
reaches  its  greatest  elongation,  it  will  appear,  for  some 
time,  to  coincide  with  the  spider  line,  and  then  leave 
it  in  the  opposite  direction. 

Clamp  the  horizontal  limb,  and  turn  the  telescope 
down  till  it  is  horizontal. 

Let  now  a  staff,  with  a  light  on  its  upper  end,  be 
carried  ten  or  fifteen  rods  distant,  toward  the  star,  and 
placed  so  as  to  range,  when  vertical,  with  the  vertical 
spider  line  of  the  telescope. 

Drive  a  stake  at  the  foot  of  the  staff,  and  another 
directly  under  the  instrument,  then  will  the  line  de- 
termined by  the  stakes  make  an  angle  with  the  true 
meridian,  equal  to  the  azimuth  of  the  North  star. 
The  true  meridian  will  lie  west  or  east  of  the 
line  of  stakes,  north  of  the  telescope,  according 
as  the  elongation  was  east  or  west,  and  may 
readily  be  located  by  the  instrument. 

The  location  of  the  meridian  can  be  verified 
thus: 

Let  AB  be  the   line   of  the   stakes   produced 
to  a   considerable  distance,   say  from   20  to  40      A 


VARIATION  OF  THE  NEEDLE.  273 

chains,   A   the   azimuth   angle,  AC  the  true  meridian, 
and  EC  perpendicular  to  AB. 

BC  can  be  found  from  the  formula, 
BC  =  AB  tan  A. 

Then  laying  off  BC  on  the  ground,  and  driving  a 
stake  at  (7,  the  stakes  A  and  C  will  trace  the  true 
meridian. 

Having  found  the  true  meridian,  the  variation  of 
the  needle  can  be  readily  determined  by  turning 
the  telescope  or  the  sights  of  the  compass  in  the 
direction  AC. 

Without  finding  the  true  meridian,  the  bearing  of 
AB  being  equal  to  the  known  azimuth  of  the  North 
star  at  its  greatest  elongation,  the  variation  of  the 
needle  can  be  found  by  directing  the  telescope  or  the 
sights  of  the  compass  in  the  direction  AB. 

The  following  method  may  be  resorted  to  by  the 
surveyor  who  does  not  possess  an  instrument  with 
a  telescope. 

Fix  a  plank,  firmly  level,  east  and  west,  about  three 
feet  above  the  ground;  then  take  a  board  about  six 
inches  square,  and  having  detached  one  of  the  com- 
pass sights,  fix  it  to  the  board,  at  right  angles  with 
its  upper  edge.  Drive  a  nail  obliquely  a  little  way 
into  the  board,  so  that  it  can  be  tacked  to  the  plank. 

About  fifteen  feet  north  of  the  plank  suspend  a 
plumb  line,  from  the  top  of  an  inclined  stake  of  such 
height  that  the  North  star,  when  seen  through  the 
sight  while  the  board  rests  on  the  plank,  will  appear 
about  one  foot  below  the  upper  end  of  the  plumb  line. 

Suspend  the  plumb  in  a  vessel  of  water  to  prevent 
the  line  from  vibrating,  and  let  an  assistant  hold  a 
light  near  it,  so  that  it  can  be  seen  through  the  sight, 


274  SURVEYING. 

About  half  an  hour  before  the  time  of  the  greatest 
elongation  of  the  North  star,  place  the  board  on  the 
plank,  and  slide  so  that  the  star  and  plumb  line  shall 
range  when  seen  through  the  sight.  As  the  star  ap- 
proaches its  greatest  elongation,  move  the  board  along 
the  plank  in  the  opposite  direction,  so  as  to  keep  the 
range. 

When  the  star  reaches  its  greatest  elongation,  it  will 
appear  to  keep  the  range  for  several  minutes,  then  it 
will  move  slowly  in  the  opposite  direction. 

Tack  the  board  to  the  plank,  taking  care  not  to 
change  its  position.  Then  let  a  staff  with  a  light 
on  its  top  be  placed  about  ten  rods  farther  to  the 
north,  so  as  to  range,  when  vertical,  through  the  sight, 
with  the  plumb  line. 

Drive  a  stake  at  the  foot  of  the  staff,  and  one  di- 
rectly under  the  plumb  line,  then  will  the  line  of  the 
stakes  make,  with  the  meridian,  an  angle  equal  to  the 
azimuth  of  the  North  star  at  its  greatest  elongation. 

The  true  meridian,  and  the  variation  of  the  compass, 
can  then  be  found  as  above. 


FIELD  OPERATIONS. 

275.    Finding  Corners. 

In  searching  for  a  corner,  first  seek  for  the  monu- 
ment, whether  tree,  post,  stake,  or  stone,  as  given  and 
witnessed  in  the  original  field  notes,  which,  if  found, 
must  be  considered  decisive  in  establishing  the  corner. 

If  no  monument  can  be  found,  the  corner  can  often 
be  found  by  indirect  methods,  of  which  the  following 
are  the  most  available: 


FIELD   OPERATIONS. 


275 


Thus,  if  a  monument  can 
be  found  at  each  of  the  cor- 
ners A,  0,  />,  but  not  at  B, 
find  the  corners  E  and  F,  at 
each  of  which  set  up  a  flag- 
staff or  high  pole,  and  send 
the  flag-man  as  near  to  B  as 
possible,  and  let  him  stand 
facing  D,  so  that  he  can  see 


signals  made  both  at  A  and  C.    .0 


The  observer  at  A  can,  by  waving  his  hand,  bring 
the  flag-man  in  the  line  AE,  and  the  observer  at  C 
can  bring  him  in  the  line  CF,  and  being  in  both  lines, 
AE  and  OF,  at  the  same  time,  he  will  be  at  their  in- 
tersection B,  the  corner  required. 

If  the  corner  E  can  be  found,  but  not  F,  measure 
AB  the  required  distance  in  the  line  AE.  If  the  dis- 
tance AB  is  not  known,  but  it  is  simply  known  that 
AB  is  equal  to  DC,  first  measure  DC.  If  neither  E  nor 
F  can  be  found,  run  AB  parallel  to  DC,  and  CB  parallel 
to  DA,  and  the  intersection  of  these  lines  will  determine 
B,  if  the  field  is  a  parallelogram. 

If  the  field  is  not  a  parallelogram,  retrace  one  of  the 
lines  terminated  lay  known  corners,  and  compare  the 
bearing  with  the  bearing  in  the  original  notes,  which 
will  give  the  variation  of  the  needle.  Then  run  the 
lines  AB  and  CB  from  the  notes,  allowing  for  the  vari- 
ation, and  the  intersection  will  determine  B. 

In  like  manner  two  or  more  lost  corners  may  be  found. 

If  the  bearings  and  distances  are  given  in  the  origi- 
nal notes,  and  but  one  corner  can  be  found,  retrace 
some  established  line  in  the  neighborhood  to  find  the 
variation,  and,  beginning  at  the  known  corner,  run  the 
lines  from  the  notes,  allowing  for  the  variation. 


276  .SURVEYING. 

The  importance  of  allowing  for  the  variation  may 
be  illustrated  thus: 

Let  the  full  lines  bound  the  lot. 

If  the  surveyor  should  run  this  lot 
from  the  original  notes,  one  corner 
being  known,  the  dotted  lines  would 
mark  the  boundaries  as  run,  and  their  intersections 
the  corners,  thus  encroaching  on  one  side,  and  leaving 
gaps  on  the  other,  which  of  course  would  never  do. 

276.  Finding  Bearings  and  Distances. 

After  finding  the  corners,  set  a  stake  at  each,  and, 
beginning  at  any  corner,  place  the  compass  or  transit 
directly  over  the  stake,  and  send  the  flag-man  to  the 
next  corner,  who  must  place  the  flag-staff'  on  the  stake. 

Take  the  bearing,  and  measure  the  distance  as  here- 
tofore directed;  and,  in  like  manner,  find  the  bearings 
and  distances  of  the  remaining  sides. 

If  obstacles  should  prevent  the  taking  of  the  bear- 
ing of  any  line,  measure  the  same  distance  from  each 
corner,  at  right  angles  to  the  line,  on  the  same  side, 
so  as  to  secure  a  line  free  from  obstacles,  and  take  the 
bearing  of  this  line,  which  will  be  the  bearing  of  the 
required  line,  since  they  are  parallel. 

Lines  are  measured  a  little  to  one  side  when  fences, 
ponds,  or  other  obstacles,  are  in  the  line. 

Thus,  if  the   perpendiculars 
AC  and  BD  are  equal, 


AB  can  be  found  by  Trigo- 
nometry,  if  AE  and  EB  and 
two  angles  be  measured. 


FIELD   OPERATIONS. 


277 


277.    Offsets. 

Offsets  are  perpendiculars  measured  from  a  line  to 
the  angles  of  a  neighboring  broken  line,  or  to  the 
banks  or  centers  of  creeks,  rivers,  or  other  bodies  of 
^vater.  Thus,  a,  6,  c. 


278.   Taking  Field  Notes. 


First  Method. 


Second  Method. 


Sta, 

Bearings. 

Dist. 

1 

N.  20°  E. 

15.50 

2 

E. 

18.00 

3 

S.  20°  E.        30,00 

4       i          W. 

25.00 

O 

, 

X.  32J°  W. 

16.09 

The  first  method  is  in  the  proper  form  for  calcula- 
tion, and  may  be  conveniently  employed  when  it  is 
not  important  to  make  a  map  of  the  lot  surveyed. 

The  second  method,  being  a  random  outline  with 
bearings  and  distances  indicated,  may  be  employed 
when  it  is  desirable  for  the  surveyor  to  keep  before 
him,  while  at  work,  an  outline  of  the  lot. 


Third  Method. 

68.00 

Station  A. 

57.60 

Orchard  fence 

42.00 

Oatfield  fence 

26.00 

Meadow  fence 

14.00 

S.  Bank  of  Greek 

13.20 

INT.  Bank  of  Creek: 

10.80 

Pasture  fence 

Station  E 

4.80 

A 

S.  Bank  of  River 
S.           Left  Bank  of  River 

18.40 

^%v     IS".  Bank  of  River 

17.40 

__9_ep._1|  Offset 

16.40 

_?-*5of  Offset 

10.50 

Station  D 

As 

S.  32°E.  Left  Bank  of  River 

f~ 

30.00 

Lei't  Bank  of  River 
NNN^      RigTit  Bank  of  River 

tfL___!L.._ 

26.00 

\Offset 

C- 

16.40 
7.30 

Orfset 
OffsetNx 

Station  C 

4.80 
A 

N.  Line  ot'SRoad 
JST.  63°E.  Right  Bank^f  B,iver 

68.00 
58.00 

Road  East              x^  j 
*.  _        "Woods 

^ 

55.20 

Pond 

42.00 

Pasture  fence-"'         ^ 

26.00 

Cornfield  fence    /^' 

10.52 

"  Wheatfield  fence 

Station  B 

A"" 

IN.  Middle  of  Turnpike 

40.00 

Lot  Line  . 

31.20 

Meadow  fence 

24.00 

/  Grove  fence 

Station  A 

17.20 
10.08/x 

A'" 

X)ooryard  fence 
Orchard  fence 
"W.  ^fiddle  of  Turnpike 

.4- 


(278) 


MAP  OF  FARM 

Scale  16  p.  to  1  inch. 


A 


(279) 


280  SURVEYING. 

279.    Remarks  on  the  Third  Method. 

The  third  method  should  be  employed  whenever  a 
map,  more  or  less  perfect,  is  to  be  made.  The  notes 
should  be  placed  on  a  left-hand  page  of  the  field  book, 
and  the  map  on  the  right  page,  facing. 

By  referring  to  the  notes  and  map  illustrating  this 
method,  it  will  be  observed  that  the  survey  began  at 
A,  the  S.  E.  corner  of  the -farm,  at  the  middle  of  the 
turnpike,  and  that  we  commenced  to  record  the  notes 
at  the  bottom  of  the  page.  . 

This  will  keep  the  notes  of  the  objects,  at  the  right 
or  left  of  each  line  run,  in  their  natural  position  on 
the  page,  at  the  right  or  left  of  the  parallel  lines  in- 
closing the  distance  from  the  station  at  the  beginning 
of  the  line  to  the  objects  worthy  of  record  encountered 
in  running  the  line. 

The  character  /\  denotes  station,  at  the  left  of  which 
stands  the  letter  marking  its  position  on  the  map,  and 
at  the  right  the  bearing  of  the  next  course. 

A  prominent  object,  such  as  the  chimney  of  the 
house,  a  large  tree  standing  in  an  open  field,  may  be 
selected,  and  its  bearings  from  the  principal  stations 
be  taken.  These  bearings  will  serve  as  checks  against 
errors  in  drawing  the  map,  and  may  aid  in  finding 
the  corners  should  they  be  lost.  In  the  present  example, 
a  chestnut  tree  on  the  top  of  a  hill,  in  the  pasture  at 
the  left  of  the  lane,  is  selected,  and  its  bearing  from 

A,  B,  and  D  given. 

0 

280.    Surveying  Creeks  and  Roads. 

1.  Creeks  may  be  meandered  as  described  under  the 
head  of  Survey  of  the  Public  Lands. 


FIELD   OPERATIONS. 


281 


2.  They  may  also  be  surveyed  by  running  straight 
lines  connecting  points  on  the  bank,  taking  the  bear- 
ings of  these  lines,  the  distances  from  the  origin  of 
these  lines  to  the  perpendicular  offsets  run  from  the 
lines  to  the  bank  of  the  river,  and  the  length  of 
the  offsets,  as  exhibited  in  the  following  field  notes 
and  plot. 


Field  Notes. 


Plot. 


Station  C  /\ 
3.48 
3.04 

Station  B  A 
6.19 
4.39 
3.14 
2.84 
2.24 
1.08 
.40 

A 


The  name  of  stations  and  the 
left-hand  offsets  are  noted  on  the 
left  of  the  parallels,  the  right- 
hand  offsets  and  bearings  on  the 
right,  the  distance  from  the  station  to  the  offsets,  and 
the  sign  for  station,  between  the  parallels. 

3.  In  surveying  an  existing  winding  road,  keep  in 
the  road,  run  straight  lines  as  far  as  possible,  without 
running  out  of  the  road,  note  the  bearing  of  these  lines, 
the  distances  to  the  offsets  at  different  points  to  the 
sides  of  the  road,  the  lengths  of  these  offsets,  and  make 
an  accurate  plot  of  the  road. 


4.  To  survey  a   new  road,   find   the   bearing   of   the 
middle  line  from   the  origin  to  the  next  angle  or  in- 
tersection with   another  road,  measuring   the  distance 
S.  N.  24. 


282  SURVEYING. 

from    the    origin    to    the    lines    of    farms,   creeks,   etc., 
which  it  intersects. 

Set  temporary  stakes  at  the  angles,  and  at  convenient 
.distances  along  the  middle  line,  to  guide  in  making  the 
'road,  and  plant  monuments  at  a  given  distance  and 
bearing  from  the  angular  points,  so  that  they  will  not 
be  disturbed  in  making  or  working  the  road.  Take 
notes,  and  make  a  correct  plot  of  the  road. 

281.  Surveying  Towns. 

Commence  at  the  intersection  of  principal  streets, 
take  their  bearings,  measure  their  lengths,  noting  the 
distances  to  the  streets  and  alleys  crossed,  taking  off- 
sets to  corners  of  streets  and  prominent  objects,  as 
public  buildings,  etc.,  till  a  prominent  cross-street  is 
reached,  which  survey  in  the  same  manner,  changing 
the  courses  at  such  stations  as  will  lead  back  to  the 
original  station. 

Survey  all  the  streets  and  alleys  enclosed.  Then  sur- 
vey an  adjoining  district,  and  so  on,  till  the  entire  town 
or  city  has  been  surveyed. 

Take  notes,  and  make  an  accurate  map  of  the  town, 
on  which  locate  not  only  the  streets  and  alleys,  but 
public  buildings,  parks,  fountains,  monuments,  etc. 

282.  Reverse  Bearing. 

Let  AB  be  a  line  run  from  A  to  B,  AN 
and  BS  meridians,  then  will  NAB  be  the 
bearing  of  A.B,  and  SBA  will  be  the  reverse 
bearing. 

Since  the  meridians  AN  and  BS  may  be 
regarded  as  parallel,  the  bearing  and  reverse 


FIELD  OPERATIONS.  283 

bearing    are    equal.     Thus,    if    the    bearing    of    AB    is 
N.  30°  E.,  the  reverse  bearing  is  S.  30°  W. 

The  bearing  and  reverse  bearing  agree  in  the  value 
of  the  angle,  and  differ  in  both  the  letters  which  in- 
dicate the  general  direction  of  the  line.  In  fact,  the 
reverse  bearing  of  a  line  is  the  bearing  of  the  line  if 
run  in  the  opposite  direction.  Thus,  SBA,  the  reverse 
bearing  of  the  line  AB,  run  from  A  to  B,  is  the  bear- 
ing of  the  line  BA,  run  from  B  to  A. 

Of  the  letters  used  in  bearings,  we  shall  call  N  and 
S  latitude  letters,  and  E  and  W  departure  letters. 

To  guard  against  inaccurate  observations,  and  the 
disturbance  of  the  needle  occasioned  by  local  attraction, 
the  reverse  bearing  should  be  taken  at  every  station. 
If  the  bearing  and  reverse  bearing  agree  in  value,  the 
bearing  may  be  considered  as  correctly  taken;  if  they 
differ  materially,  both  should  be  taken  again.  If  they 
still  differ,  the  difference  may  be  regarded  as  occasioned 
by  local  attraction. 

To  ascertain  at  which  station  the  local  attraction 
exists,  place  the  instrument  at  a  third  station,  at  a 
considerable  distance  from  each  of  the  doubtful  stations, 
and  sight  to  each,  then  from  these  back  to  the  third 
station.  The  local  attraction  may  be  considered  to  exist 
at  the  station  where  the  bearing  of  the  third  station 
disagrees  with  its  bearing  taken  at  the  third  station. 

If  the  error  occurred  in  the  foresight,  correct  it  before 
entering  the  bearing  in  the  field  notes,  and  note  the 
amount  of  disturbance;  if  the  error  occurred  in  the 
backsight,  the  next  foresight  will  be  affected,  and  should 
be  corrected  before  entered. 


284 


SURVEYING. 


PRELIMINARY   CALCULATIONS. 
283.    Angles  between  Courses. 

1.  If  the  latitude  letters  are  alike,  also  the  departure  letters, 
the  included  angle  is  equal  to  the  difference  of  the  bearings. 

If  AB  bears  N.  40°  E.,  and  AC 
N.  20°  E.,  BAG  =  BAN— CAN  =  40° 

-  20°  =  20°. 

If  AD  bears  S.  40°  W.,  and  AE 
S.  20°  W.,  DAE  =  DAS  —  EAS  =  40° 

-  20°  =  20°. 

2.  If  the  latitude  letters  are  alike,  and  the  departure  letters 
unlike,  the  included  angle  is  equal  to  the  sum  of  the  bearings. 

If  AB  bears  N.  38°  E.,  and  AC 
N.  18°  W.,  BAC=BAN+NAC=S&° 
H-  18°  =  56°. 

If  AD  bears  S.  38°  W.,  and  AE 
S.  18°  E.,  DAE  =>  DAS  +  SAE  =  38° 
+  18°  =  56°. 

3. .  If  the  latitude  letters  are  unlike,  and  the  departure  letters 
alike,  the  included  angle  is  equal  to  180°  minus  the  sum  of 
the  bearings. 

If  AB  bears  N.  45°  E.,  and  AE 
S.  30°  E.,  BAE  =  180°—  (NAB  +SAE) 
=  180°  —  75°=  105°. 

If  AD  bears  S.  45°  W.,  and  AC 
=  180°  —  75°  =  105°. 


PRELIMINARY  CALCULATIONS. 


285 


4.  If  the  latitude  letters  are  unlike,  also  the  departure  letters, 
the  included  angle  is  equal  to  180°  minus  the  difference  of  the 


N    E 

If  AB  bears  N.  45°  E.,  and  AC 
S.  15°  W.,  BAG  =  180°  —  (NAB  — 
SAC)  =  180°  —  30°  =  150°. 

If  AD   bears   S.   45°    W.,    and    AE 
N.    15°    E.,    DAE  ±=  180°  —  (SAD  - 
NAE)  =  180°  —  30°  ==  150°. 

Remark.— These  principles  apply  when  both  courses 
run  from  or  toward  the  vertex ;  if  one  runs  from  the 
vertex,  and  the  other  toward  it,  reverse  the  bearing  of 
one  side  before  applying  the  principles. 


c   s 


284.    Examples. 

1.  Find  the  angle  A,  if  AB  bears  N.  78°  E.,  and  AC 
N.  24°'  E.  Ans.  54°. 

2.  Find  the  angle  A,  if  BA  bears  S.  34°  E.,  and  AC 
S.  48°  W.  Ans.  98°. 

3.  Find  the  angle  A,  if  BA  bears  S.  70°  W.,  and  CA 
N.  25°  E.  Ana.  135°. 

4.  Find  the   angles  of  the   polygon  ABODE,  if  AB 
bears  N.  30°  E. ;   BC,  N.  60°  E. ;    CD,  S.  50°  E.;  DE,  S. 
40°  W. ;   EA,  N.  78°  W. 

,4  =  72°,   5=150°,   (7=110°,   £>  =  90°,  JE=118°. 


285.   Problem. 

Given  the  bearings  of  the  sides  of  a  field,  to  find  the  bear- 
ings if  the  field  be  supposed  to  revolve,  so  as  to  cause  one  of 
the  sides  to  become  a  meridian. 


286 


SURVEYING. 


In  the  following  diagram  let  the  full  lines  denote 
the  original  position  of  the  sides  of  the  field,  a  the 
side  that  is  to  become  the  meridian,  and  the  dotted 
lines  the  revolved  position  of  the  sides. 


a,  N.  30°  E. 
6,  N.  60°  E. 

c,  N.  10°  E. 

d,  S.-  45°  E. 

e,  S.   75°  E. 
/,      S. 

g,  S.  55°  W. 
h,  S.  20°  W. 
t,       W. 
j,  N.  25°  W. 
'fc,  N.  80°  W. 


a',  N. 
6',  N.  30°  E. 
•c'.  X.  20°  W. 
d',  S.  75°  E. 
e',  X.  75°  E. 
/',  S.  30°  E. 
#',  S.  25°  W. 
tf,  S.  10°  E. 
*',  S.  60°  W. 
/,  X.  55°  W. 
F,  S.  70°  W. 


From  the  above  illustration  we  derive  the  following 
principles :" 

1.  If  the  letters  which  indicate  the  general  direction 
of  the  side  which   is   to  be   made  a  meridian  are  both 
alike  or  both  unlike  those  of  another  side,  then, 

1st.  If  the  bearing  of  the  former  is  less  than  that  of 
the  latter,  the  difference  of  the  bearings  will  be  the 
bearing  of  the  latter,  the  letters  remaining  the  same 
as  before. 

2cL  If  the  bearing  of  the  former  is  greater  than  that 
of  the  latter,  the  difference  of  the  bearings  will  be  the 
bearing  of  the  latter,  the  departure  letter  being 
changed. 

2.  If  one   of  the  letters  which   indicate   the   general 
direction  of  the  side  which  is  to  be  made  a  meridian 
is  like  and  the  other  unlike   the   corresponding   letter 
of  another  side,  then, 


PRELIMINARY  CALCULATIONS.  287 

1st.  The  sum  of  the  bearings,  if  less  than  90°,  will 
be  the  bearing  of  that  side,  the  letters  remaining  -the 
same  as  before. 

2cL  If  the  sum  of  the  bearings  is  greater  than  90°, 
its  supplement  will  be  the  bearing  of  that  side,  the 
latitude  letter  being  changed. 

286.    Examples. 

1-.  The  bearings  of  the  sides  of  a  field  are  as  follows : 

1st,  N.  30°  E. ;    2d,  N.  60°  E. ;    3d,  S.  40°  E. ;    4th,  S. 

30°  W.;    5th,  W.;    6th,  N.  18J°  W.    Find  the  bearings 

of  the  sides  if  the  second  side  becomes  a  meridian. 

Ans.  1st,  N.  30°  W. ;    2d,  N. ;    3d,  N.  80°  E.;   4th,  S. 

30°  E.;    5th,  S.  30°  W. ;    6th,  N.  78f°  W.  <? 

Ov-* 

2.  The  bearings  of  the  sides  of  a  field  are  as  follows  : 

1st,  N.  45°  W.;  2d,  N.  18°  E.;  3d,  E.;  4th,  N.  32°  E.; 
5th,  S.  42^°  E.;  6th,  S.;  7th,  S.  65J°  W.  Find  the 
bearings  if  the  first  side  be  made  a  meridian. 

Ans.  1st,  N.;  2d,  N.  63°  E.;  3d,  S.  45°  E.;  4th,  N.  77° 
E.;  5th,  S.  2|°  W. ;  6th,  S.  45°  W. ;  7th,  N.  69f°  W. 

3.  The  bearings  of  the  sides  of  a  field  are  as  follows  : 
1st,  N.  20°  E.;    2d,  N.  70°  E. ;    3d,  E.;    4th,  S.  45°  E.  ; 

5th,  S. ;    6th,  S.  45°  W. ;    7th,  W. ;    8th,  N.  3|°  W.    Find 
the  bearings  if  the  sixth  side  be  made  a  meridian. 

Ans.  1st,  N.  25°  W. ;  2d,  N.  25°  E. ;  3d,  N.  45°  E. ; 
4th,  E.;  5th,  S.  45°  E. ;  6th,  S. ;  7th,  S.  45°  W.;  8th, 
N.  48f°  W. 

287.   Latitude  and  Departure. 

The  latitude  of  a  course  is  the  distance  between  the 
two  parallels  of  latitude  passing  through  the  extremi- 
ties of  the  course. 


288 


SURVEYINQ. 


The  departure  of  a  course  is  the  distance  between 
the  two  meridians  passing  through  the  extremities  of 
the  course. 

Let  AB  be  a  course,  AD  and  BC  paral- 
lels of  latitude,  and  ^Cand  BD  meridians. 
Then  will  AC  or  DB  be  the  latitude  of  the 
course,  and  CB  or  AD  its  departure. 

But  AC  =  AB  X  cos  CAB, 
and  CB  =  AB  X  sin  CAB. 

Hence,  latitude  =  course  X  cosine  of  bearing, 
and    departure  —  course  X  sine  of  bearing. 

If  the  line  runs  due  east  or  west,  its  latitude  is  0. 

If  the  line  runs  due  north  or  south  its  departure  is  0. 

Latitude  north  is  considered  plus;  latitude  south,  minus. 

Departure  east  is  considered  plus;  departure  west,  minus. 

For  brevity  let  us  designate  the  bearing  by  b,  the 
course  by  c,  the  latitude  by  I,  and  departure  by  d,  then 
we  shall  have  the  cases  given  in  the  following  article: 

288.   Table  of  Cases. 


Given. 

Req. 

Formulas. 

1 

b,  r, 

l,d. 

I  —  c  cos  b,        d  =  c  pin  ft. 

2 

M, 

c,d. 

r                                       rl         1  tan   A 

cos  b 

3 

b,d, 

c,l. 

c        d               I          d 

sin  6'                   tan  b 

4 
5 

c,  I, 
c,d, 

b,d. 

cos  6=—,               d=  i/c2  —  P. 

C 

sin  6  =  —  ,                /  =  1/c2  —  d2. 

6 

w 

b,c. 

^                            _ 
^ 

PRELIMINARY  CALCULATIONS.  289 

289.   Examples. 

1.  Given  b  =  N.  53°  20'  E.,  and  c  ==  26.50  ch.  ;  required 

I  and  d.  Ans.  I  =  15.82  ch.  N.,  d  =  21.26  ch.  E. 

2.  Given  b  =  S.  75°  47'  W.,  and  /  =  22.04  ch.  S.  ;  re- 
quired c  and  d.  Ans.  c  =  89.75  ch.,  d  ==--  87  ch.  W. 

3.  Given  b  =  N.  35°  W.,  and  d  -  1.55  ch.W.  ;  required 
c  and  /.  Ans.  c  ==  2.70  ch.,  J  ==  2.21  ch.  N. 

4.  Given  c  *±  35.35  ch.,.  and  I  =  31  ch.  N.  ;    required 
b  and  d. 

Ans.  b  =  N.  28°  44'  E.  or  W.,  d  ==  16.99  ch.  E.  or  W. 

5.  Given  c  =  31.30  ch.,  and  d  as  22.89  ch.W.  ;  required 
b  and  £. 

6  ==  N.  or  S.  47°  W.,  and  I  =  21.35  ch.  N.  or  S. 


6.  Given  I  =  7.02  ch.  S.,  and  d  =  7.14  ch.W.  ;  required 
b  and  c.  Ans.  b  =  S.  45°  29'  W.,  c  ==  10.01  ch. 


290.   Traverse  Table. 

The  traverse  table  affords  a  ready  method  of  finding 
the  latitude  and  departure  of  a  course  whose  distance 
and  bearing  are  given. 

Let  us  find  the  I  and  d  of  a  line  whose  b  is  N.  35° 
15'  E.,  and  c  =  47.85  ch. 

Turning  to  the  traverse  table,  under  35°  15'  we  find 

c  =--  40*  gives  I  =  32.67,  d  =  23.09. 

c  =       7  gives  I  --=    5.72,  d  =    4.04. 

c=  .8  gives  I  =       .65,  d  =      .46. 

c  ==  .05  gives  1=      .04,  d  =      .03. 

.  • .    c  ^  47.85  gives  I  =  39.08,  d  =  27.62. 

S.  N.  25. 


290  SURVEYING. 

The  I  and  d  for  40  are  found  from  the  I  and  d  of  4, 
as  given  in  the  table,  by  multiplying  by  10,  or  remov- 
ing the  decimal  point  one  place  to  the  right. 

The  I  and  d  for  the  distance  7  are  given  in  the  table, 
but  the  right  hand  figure  is  dropped,  and  1  is  carried 
if  the  figure  dropped  exceeds  5. 

The  I  and  d  for  the  distance  .8  are  found  from  the 
I  and  d  for  the  distance  8  by  removing  the  decimal 
point  one  place  to  the  left,  rejecting  the  figures  at  the 
right  of  the  second  decimal  place,  carrying  as  above. 

For  the  distance  .05,  remove  the  decimal  point  two 
places  to  the  left,  reject  and  carry  as  before. 

If  the  bearing  exceeds  45°,  the  I  and  d  will  be  found 
in  columns  marked  at  the  bottom  of  the  page. 

291.  Examples. 

1.  Given  b  =  N.  28°  45'  E.,  and  c  =  35.35  ch. ;  required 
I  and  d.  Ans.  I  ==  30.98  ch.  N.,  d  ==  17  ch.  E. 

2.  Given  b  =  S.  36f  °  E.,  and  c=  19.36  ch. ;   required  I 
and  d.  Am.  I  =  15.51  ch.  S.,  d  =  11.59  ch.  E. 

3.  Given   b  =  N.   53°  15'  E.,  c  =  11.60  ch.;    required 
I  and  d.  Am.  I  ~  6.94  ch.  N.,  d  =  9.29  ch.  E. 

4.  Given  b  =  S.  74i°  E.,  c  =  30.95  ch. ;  required  I  and  c?. 

Ans.  I  =  8.27  ch.  S.,  d  =  29.83  ch.  E. 

5.  Given  b  =  N.  33J°  W.,  c  =  37  ch. ;  required  I  and  d. 

Ans.  I  ===  30.94  ch.  N.,  d  =  20.29  ch. W. 

6.  Find   the  Z  and  d  of  the   sides  of  a  lot  of  which 
the  following  are  the  field  notes:    Commencing  at  the 
most  westerly  station,  and  running   thence   N.  52°  E., 
21.28  ch.;    thence  S.  29f°  E.,  8.18  ch.;    thence  S.  31f° 
W.,  15.36  ch.;  thence  N.  61°  W.,  14.48  ch.,  to  the  point 
of  beginning. 


PRELIMINARY  CALCULATIONS. 
The  work  is  written  thus: 


291 


Sta. 

Bearings. 

Dist, 

tf.  £a*. 

,9.  Lat. 

£.  Dep. 

W.Dep. 

1 

N.  52°  E. 

21.28 

13.10 

16.77 

2 

S.  29f°  E. 

8.18 

7.11 

4.06 

r> 

I 

S.  31f°W. 
N.  61°  W. 

15.36 
14.48 

7.02 

13.06 

8.08 
12.67 

292.    Balancing  the  Work. 

It  is  evident  that  in  passing  around  a  field  to  the 
point  of  beginning,  we  have  gone  just  as  far  north  as 
south,  and  just  as  far  east  as  west.  Hence,  the  sum 
of  the  northings  should  be  equal  to  the  sum  of  the 
southings,  and  the  sum  of  the  eastings  to  the  sum  of 
the  westings. 

In  practice,  however,  this  is  seldom  the  case,  owing 
to  the  fact  that  the  bearings  are  taken  only  to  quarter 
degrees,  and  that  the  chaining  is  not  perfectly  correct. 

It  is  not  a  settled  point  among  surveyors  how  great 
an  error  in  latitude  or  departure  can  be  allowed  with- 
out resurveying  the  lot.  Some  would  admit  an  error 
of  1  link  for  every  10  chains  in  the  sum  of  the  courses; 
others,  1  link  for  every  3  chains.  Each  surveyor  must 
settle  this  point  for  himself  by  ascertaining,  by  expe- 
rience, how  nearly  he  can  make  his  work  balance. 

When  an  error  is  as  likely  to  occur  in  one  course 
as  in  another,  the  errors  of  latitude  and  departure  are 
distributed  among  the  courses  in  proportion  to  their 
length. 

It  will  not,  in  general,  be  necessary  to  make  all  the 
proportions,  for  after  making  one  for  latitude  and  one 
for  departure,  the  remaining  corrections  can  be  made 
by  a  comparison  of  distances. 


292 


SURVEYING. 


Let  us  take  example  6  of  the  last  article. 


StoJ  Bearings. 

DM. 

SLat. 

SLat. 

EDep. 

}\T)<p. 

CIV/,. 

CSX. 

CED. 

CWD. 

\ 
1  j  N.52°E. 

21.28 

13.10 

16.77 

13.12 

16.74 

2  j  S.29J°E. 

8.18 

7.11 

4.06 

7.10 

4.05 

3    S.SlfW. 

15.36 

13.06 

8.08 

13.05 

8.1C 

4  JN.61°\V. 

14.48 

7.02 

12.67 

7.03 

12.69 

59.30 

20.12 

20.17 

20.83 

20.75 

20.15 

20.15  120.79 

20.79 

Error  in  Lat.  = 
Error  in  Dep.  = 

Corrections  for  latitude. 

59.30  :  21.28  : :  .05  :  .02. 

59.30  :    8.18  ::  .05  :  .01. 

59.30  :  15.36  : :  .05  :  .01. 

59.30  :  14.48  : :  .05  :  .01. 


20.17  — 20.12  =  .05. 
20.83  —  20.75  =  .  08. 

Corrections  for  Departure. 

59.30  :  21.28  : :  .08  :  .03. 

59.30  :    8.18  ::  .08  :  .01. 

59.30  :  15.36  : :  .08  :  .02. 

59.30  :  14.48  : :  .08  :  .02. 


The  corrections  are  made  to  the  nearest  link  or 
hundredth. 

Since  the  north  latitude  is  too  small,  and  the  south 
latitude  too  great,  add  to  each  north  latitude  the  corre- 
sponding correction,  and  subtract  from  the  south  lati- 
tude. In  a  similar  manner  correct  the  departure. 

If  one  side  is  much  more  difficult  to  measure  than 
the  remaining  sides,  it  is  to  be  presumed  that  the  error 
occurred  chiefly  in  measuring  that  side,  and  the  correc- 
tions should  be  made  accordingly. 

If,  in  taking  one  bearing,  the  object  could  not  be 
distinctly  seen,  the  error  probably  occurred  in  that 
bearing;  then  correct  mainly  in  the  latitude  and  de- 
parture of  that  course. 

In  practice  it  will  not  be  necessary  to  make  addi- 
tional columns  for  the  corrected  latitude  and  departure, 
since  they  may  be  written  in  the  same  columns,  over 
the  others,  with  different  colored  ink. 


PRELIMINARY  CALCULATIONS.  293 

293.   Examples. 

1.  Find  the  I  and  d,  and  balance  the  work  from  the 
following  notes: 

1st,  N.  34i°  E.,  8.19  ch.;  2d,  N.  85°  E.,  3.84  ch.;  3d, 
S.  56f°  E.,  6.60  ch.;  4th,  S.  34J°  W.,  10.59  ch.;  5th, 
i\.  56°  W.,  9.60  ch. 

2.  Find  the  I  and  <i,  and  balance  the  work  from  the 
following  notes : 

1st,  N.  5°  E.,  22.50  ch.;  2d,  S.  83°  E.,  12.96  ch.;  3d, 
N.  50°  E.,  19.20  ch.;  4th,  S.  32°  E.,  32.76  ch.;  5th,  S. 
41°  W.,  12.60  ch.;  6th,  W.,  16.86  ch.;  7th,  N.  79°  W., 
21.84  ch. 

3.  Find  the  balanced  I  and  d  of  the  following: 

1st,  N.  30°  E.,  10  ch.;  2d,  N.  60°  E.,  18.18  ch. ;  3d, 
S.  40°  E.,  20.10  ch.;  4th,  S.  30°  W.,  24.50  ch. ;  5th,  W, 
15  ch. ;  6th,  N.  18|°  W.,  19.92  ch. 

294.   Double  Meridian  Distance. 

The  double  meridian  distance  of  a  course  is  double  the 
distance  of  its  middle  point  from  a  given  meridian. 

Let  AB  be  a  given  course,  NS  the  given 
meridian,  P  the  middle  point  of  AB,  PQ 
perpendicular  to  NS. 

Then  will  2  QP  be   the   double  meridian    Q " 
distance  of  AB. 

In  the  following  illustration  we  shall  as- 
sume that  the  meridian  of  reference  passes 
through  the  most  westerly  station,  which  we  shall  call 
the  principal  station,  that  departures  east  are  plus,  and 
west,  minus,  that   the  lines  were -run   in  the  direction 


294 


SURVEYING. 


ABCD,  so  as  to  keep  the  field  on        ,  B 

E 


the  right. 

The  following  relations  can  be     A 
verified  from  the  diagram:  T 


Q  ...... 


1. 

3.  2VU=2TR  +  FC-}-  (—  GD).     v 

4.  2XW=2VU+(—GD) 

>•  x  _H. 


1.  TTie  double  meridian  distance  of  the  first  course  is  equal 
to  its  departure. 

2.  The  double  meridian  distance  of  the  second  course  is 
equal  to  the  double  meridian  distance  of  the  first  course,  plus 
the  departure  of  the  first  course,  plus  the  departure  of  the 
second  course. 

3.  The  double  meridian  distance  of  any  course  is  equal  to 
the  double  meridian  distance  of  the  preceding  course,  plus  the 
departure  of  that  course,  plus  the  departure  of  the  given  course. 

4.  The  double  meridian  distance  of  the  last  course  is  equal 
to  its  departure  with  its  sign  changed. 

Take  the  example  of  a  preceding  article,  as  balanced. 


Sta. 

Bearings. 

DM. 

NLat. 

SLat. 

EDep. 

WDep. 

DMD. 

1 

N.52°E. 

21.28 

13.12 

16.74 

16.74 

2 

S.  29|°  E. 

8.18 

7.10 

4.05 

37.53 

3 

S.31f°W. 

15.36 

13.05 

8.10 

33.48 

4 

N.61°W. 

14.48 

7.03 

12.69 

12.69 

Dep.  of  1st  course  — 
-f  dep.  of  1st  course  = 
-{-  dep.  of  2d  course  = 


16.74  =  D.M.D.  of  1st  course. 
16.74 
4.05 


37.53  =  D.M.D.  of  2d  course. 


PRELIMINARY  CALCULATIONS.  295 

-f  dep.  of  2d  course  &       4.05 

41.58 
-j-  dep.  of  3d  course  =  -  -   8.10 

33.48  =  D.M.D.  of  3d  course. 
-f  dep.  of  3d  course  =  —   8.10 

25.38 
-f  dep.  of  4th  course  ==  — 12.69 

12.69  =  D.M.D.  of  4th  course. 

The  principal  or  most  westerly  station  is  not  always 
the  first  station  in  the  field  notes. 

It  will  be  observed  that  the  word  plus,  in  the  above 
principles  and  illustrations,  is  used  in  the  algebraic 
sense,  that  east  departure  is  considered  plus  and  west 
departure  minus]  that  plus,  an  east  departure,  is  a  plus 
quantity,  and  plus  a  west  departure  a  minus  quantity ; 
and  that  the  double  meridian  distance  of  the  last  course 
is  equal  to  its  departure  with  its  sign  changed,  which 
will  serve  as  a  verification  of  the  work. 

The  first  station  of  the  notes,  in  the  preceding  ex- 
ample, is  the  most  westerly,  and  was  therefore  taken  for 
the  principal  station. 

The  most  westerly  station  can  readily  be  determined 
by  inspecting  the  bearings  of  the  courses  as  given  in 
the  field  notes,  and  should  be  taken  as  the  principal 
station,  and  the  corresponding  course  as  the  first  course 
in  finding  the  double  meridian  distances. 

295.    Examples. 

1.  Given  the  following  field  notes  : 

1st,  N.  30°  E.,  10  ch.;  2d,  N.  60°  E.,  18.18  ch. ;  3d, 
S.  40°  E.,  20.10  ch,;  4th,  S.  30°  W.,  24.50  ch. ;  5th,  W., 
15  ch. ;  6th,  N.  18°  45'  W.,  19.92  ch. :  Required  the 


296  SURVEYING. 

latitude  and  departure ;  balance  the  work,  and  find  the 
double  meridian  distances. 

2.  Given  the  following  field  notes : 

1st,  N.  45°  W.,  20  ch.;  2d,  N.  18°  E.,  12.25  ch. ;  3d, 
E.,  12.80  ch.;  4th,  N.  32°  E.,  6.50  ch. ;  5th,  S.  42J0  E., 
13.20  ch.;  6th,  S.,  14.75  ch.;  7th,  S.  65J°  W.,  16.30  ch. : 
Required  the  corrected  latitude  and  departure,  and  the 
double  meridian  distances. 


AREA  OF  LAND. 
296.   Table  of  Linear  Measure. 

Mi.    Ch.       Eds. 
1  =  80=  320  = 
1  =       4  = 

I  - 


Yds. 

Ft. 

Lks. 

7n. 

1760 

=  5280 

=  8000 

=  63360. 

22 

==   66 

100 

=  792. 

5| 

•  —   16^ 

r=   25 

198. 

1 

=   3~ 

4T6, 

r=    36. 

1 

:  14": 

O  i 

I  =    12. 

1 

—    7| 

297.   Table  of  Superficial  Measure. 

Mile.       Acres.        Roods.       Chains.        Perches.  Links. 

I  —    640  =      2560  ±  -.    6400           102400  =  64000000. 

1  —          4  -         10    :             160  =  100000. 

1  =           2J  =            40  =  25000. 

1    =            16=  10000. 

1  =  625. 

Note  1. — It  should  oe  remembered  that  in  finding  the 
area  of  a  tract  of  land  the  inequalities  of  its  surface  are 
not  considered,  but  the  tract  is  treated  as  a  horizontal 
plane. 


AREA  OF  LAND. 


297 


Note  2. — The  area  of  a  portion  of  land  can,  in  a  great 
variety  of  cases,  be  calculated  by  the  rules  already  given 
for  Mensuration  of  Plane  Surfaces. 


298.    Problem. 

To  find  the  area  of  a  tract  of  land  when  the  length  and 
direction  of  the  bounding  lines  are 
given. 

It  is  evident  from  the  diagram 
that  the  area  of  A  BCD  is  equal  to 
the  sum  of  the  trapezoids  EBCY 
and  YCDH,  minus  the  sum  of  the 
triangles  AEB  and  ADH ;  and 
that  twice  the  sum  of  the  trape- 
zoids, minus  twice  the  sum  of 
the  triangles,  is  equal  to  twice 
ABCD. 

The  following  table  will  exhibit  the  general  form  of 
operation : 


Sta. 

Cour. 

NLat. 

SLat. 

DMD. 

Triangles. 

Trapezoids. 

1 

2 
3 
4 

AB 
BC 
CD 
DA 

AE 
HA 

EY 
YH 

2QP 
2TR 
2VU 
<2XW 

2QPXAE 
2XWXHA 

2TRXEX 
2VUXXH 

i 

It  will  be  observed  that  we  have  taken  the  most 
westerly  station  for  the  principal  station,  and  have 
multiplied  the  double  meridian  distance  of  each  course 
by  its  latitude,  and  that  the  product  is  double  the 
area  of  a  triangle  when  the  latitude  is  north,  and 
double  the  area  of  a  trapezoid  when  the  latitude  is 
south. 


298 


SURVEYING. 


If  we  had  taken  the  most  easterly  station  for  the 
principal  station,  the  reverse  would  be  true. 

In  the  above  we  have  supposed  that  the  lines  were 
run  in  such  direction. as  to  keep  the  lot  at  the  right. 

If  the  lines  were  run  in  the  opposite  direction,  so  as 
to  keep  the  lot  at  the  left,  the  reverse  would  be  true. 

In  any  case,  the  sum  of  the  double  areas  of  the  trape- 
zoids,  minus  the  sum  of  the  double  areas  of  the  tri- 
angles, is  equal  to  double  the  area  required. 


299.   Rule. 

Multiply  the  double  meridian  distance  of  each  course  by  its 
latitude,  placing  the  product  in  one  column  when  the  latitude 
is  north,  and  in  another  column  when  the  latitude  is  south, 
and  divide  the  difference  of  the  sums  of  the  tivo  columns  by  2, 
and  the  quotient  will  be  the  area  required. 

Take  the  example  of  a  preceding  article  whose 
D.  M.  DSs  have  been  found. 


Sta. 

Bearing*. 

Dint. 

NLat. 

SLat. 

EDep. 

WDep. 

DMD. 

Triany. 

Trait. 

1 

N.52°E. 

21.28 

13.12 

16.74 

16.74 

219.6288 

2 

S.29f°E. 

8.18 

7.10 

4.05 

37.53 

266.4630 

3 

S.312°W. 

15.36 

13.05 

8.10 

33.48 

436.9140 

4 

N.61°W. 

14.48 

7.03 

12.69 

12.69 

89.2107 

Area  =  19  A.  2  R.  36  P. 


Triangles.  Trapezoids. 

16.74X13.12  =  219.6288.        37.53  X   7.10  =  266.4630. 
12.69  X    7.03=   89.2107.        33.48X13.05  =  436.9140. 

Divide  double  the  area  by  2,  the  result 
by  10  to  reduce  the  chains  to  acres,  multi- 
ply the  decimal  by  4  to  reduce  to  roods, 
and  the  next  decimal  by  40  to  reduce  to  perches. 


308.8395  703.3770 
308.8395 
2)394.5375 
10)197.26875 
19.726875 
4 

2.907500 
40 
36.300000 


AREA  OF  LAND.  299 

300.    Plotting. 

Plotting  is  the  process  of  representing,  to  a  given 
scale,  the  length,  direction,  and  relative  position  of  the 
bounding  lines  of  a  tract  of  land. 

1st  Method. — By  means  of  latitudes  and  departures, 

Take  the  example  of  the  last  article. 

Let  NS  represent  the  meridian 
passing  through  the  principal 
station  A. 

Select  a  scale  whose  unit  shall 
represent  1  ch.,  and  take  AE  = 
13.12  ch.,  the  lat.  of  first  course. 

Through  E  draw  a  line  perpen- 
dicular to  NS;  take  EB  =  16.74 
ch.,  the  dep.  of  first  course,  and 
draw  AB. 

Through  B  draw  a  meridian,  and  take  BF  =  7.10,  the 
lat.  of  second  course. 

Through  F  draw  a  line  perpendicular  to  BF;  take 
FC  =  4.05  ch.,  the  dep.  of  second  course,  and  draw  BC. 

Through  C  draw  a  meridian,  and  take  CG  =  13.05,  the 

lat.  of  third  course. 

i 
Through  G  draw  a  line  perpendicular  to  CG,  and  take 

GD  =  8.10  ch.,  the  dep.  of  third  course,  and  draw  CD. 

Through  D  draw  a  meridian,  and  take  DI  =  7.03  ch., 
the  lat.  of  fourth  course. 

Through  /  draw  a  line  perpendicular  to  DI;  take  IA 
-  12.69  ch.,  the  dep.  of  fourth  course,  and  draw  DA. 

Remark  1.  —  If  the  departure  of  fourth  course  termi- 
nates at  A,  the  w*ork  will  be  verified. 


300 


SURVEYING. 


2.  It  will  be  observed  that  N.  lat.  is  laid  off  upward, 
S.  lat.  downward,  E.  dep.  to  the  right,  and  W.  dep.  to 
the  left. 

3.  The   auxiliary  lines  can  be  drawn  with   a   pencil 
and  afterward  erased. 

4.  If  every  scale  in  possession  of  the  surveyor  should 
make   the   diagram  too  large  or  too  small,  all  the  lati- 
tudes and  departures  can  be  divided  or  multiplied  by 
the  same    number,  and   the    results   taken    instead   of 
the  given  latitudes  and  departures. 

2d  Method. — By  means  of  bearings  and  distances. 

Take  the  same  example. 

Let  NS  represent  the  meridian 
passing  through  the  principal 
station  A. 

With  a  protractor  lay  off  the 
angle  NAB  =  52°,  the  bearing  of 
first  course,  and  take  AB  =  21.28 
ch.,  the  first  course. 

Through  B  draw  a   meridian, 
and  lay  off  &'BC—-29%°,  the  bearing  of  second  course, 
and  take  BC  =  8.18  ch.,  the  second  course. 

Through  C  draw  a  meridian,  and  lay  off  S"CD=  31f°, 
the  bearing  of  third  course,  and  take  CD  =  15.36  ch.,  the 
third  course. 

Through    D   draw   a    meridian,  and   lay  off  N'DA  = 
61°,  the  bearing  of  fourth  course,  and  take  DA  ==  14.48 
ch.,  the  fourth  course,  which  will  terminate  at  A  if  the 
work  is  correct. 

Remark  1. — The  latitude  and  departure  letters  indicate 
the  general  direction  of  the  lines,  and  the  degrees  the 
exact  direction. 


AREA  OF  LAND. 


301 


2.  Let  the  examples  of  the  following  article  be  care- 
fully plotted,  and  the  area  be  found. 

3.  By  a  careful  inspection  of  the  bearings,  the  most 
westerly   station    can    be    found,   which    take    for    the 
principal  station. 

4.  The  distances  are  all  given  in  chains. 


1. 


301.   Examples. 


Sta. 

Bearings. 

Dist. 

1 

N.  30°    E. 

10. 

2 

N.  60°   E. 

18.18 

3 

8.  40°    E. 

20.10 

4 

S.  30°    W. 

24.50 

5 

W. 

15. 

6 

N.  18|°  W. 

19.92 

2. 


Sta. 

Bearings. 

Dist. 

1 

N.  47°  E. 

15.65 

2 

S.  57°  E. 

10.55 

3 

S.  28|°W. 

17.67 

4 

S.  29J°W. 

1.11 

5 

S.  54°  W. 

1.04 

6 

N.  40J°W. 

15.90 

Ans.  80  A.  1  R.  25  P. 


Ans.  23  A.  0  R.  38  P. 


3. 


4, 


Sta. 

Bearings. 

Dist. 

1 

N.  45°  W. 

20. 

2 

N.  18°  E. 

12.25 

3 

E. 

12.80 

4 

N.  32°  E. 

6.50 

5 

S.  42i°E. 

13.20 

6 

S. 

14.75 

7 

S.  65}°  W. 

16.30 

Sta. 

Bearings. 

Dist. 

1 

N.  58°  E. 

12.97 

2 

S.  27f°E. 

3.30 

3 

S.  85J°E. 

11.65 

4 

S.  19°  E. 

15.56 

5 

S.  66i°W. 

14.03 

6 

N.  64°  W. 

14.86 

7 

N.  15i°W. 

11.23 

Ans.  58  A.  3  R.  30  P. 


Ans.  45  A.  2  R.  5  P. 


302 


SURVEYING. 


5. 


Sta. 

Bearings. 

Dist. 

1 

N.  20°  E. 

12.20 

2 

N.  70°  E. 

15.50 

3 

E. 

18.25 

4 

S.  45°  E. 

20.00 

5 

S. 

20.00 

6 

S.  45°  W. 

20.00 

7 

W. 

18.25 

8 

N.  30|°W. 

36.66 

Sta. 

Bearings. 

Dist, 

1 

S.  34°  E. 

4.56 

2 

S.  66J°W. 

13.84 

3 

N.  12J°E. 

12.15 

4 

N.  48J0W. 

12.30 

5 

N.  58f°E. 

9.92 

6 

N.  39i°E. 

5.22 

7 

S.  45i°E. 

18.63 

8 

S.  52i°W. 

10.76 

An*.  188  A.  3  R.  20  P. 


Ans.  32  A.  2  R.  26  P. 


7. 


8. 


Stfi. 

Bearings. 

Dist. 

1 

N.  30°  E. 

15. 

2 

N.  60°  E. 

15. 

3 

E. 

15. 

4 

S.  60°  E. 

15. 

5 

S.  30°  E. 

15. 

6 

S. 

15. 

7 

S.  30°  W. 

15. 

8 

S.  60°  W. 

15. 

9 

W 

15. 

10 

N.  60°  W. 

15. 

11 

N.  30°  W. 

15. 

12 

N. 

15. 

Sta. 

Bearings. 

Dist, 

1 

S.  76J°E. 

6.69 

2 

S.  14J°W. 

5.96 

3 

S.  38°  E. 

9.82 

4 

N.  30J°E. 

8.63 

5 

S.  73i°E. 

9.43 

6 

S.  !Qf°W. 

15.70 

7 

S.  42J°W. 

13.06 

8 

N.  64°  W. 

11.93 

9 

S.  79J°W. 

10.45 

10 

N.  22J°W. 

11.60 

11 

N.  37J°E. 

14.37 

12 

N.  22f  °E. 

10.79 

Ans.  251.9  A.+ 


Ans.  76.14  A.— 


AREA  OF  LAND. 


303 


302.   Problem. 

To  find  the  area  when  offsets  are  taken. 

Find  the  area  of  the 
tract  of  land  bounded 
by  the  full  lines  and 
middle  of  the  river,  as 
shown  in  the  annexed 
diagram. 

Having  run  the  sta- 
tionary line  CD,  we  have 
the  following  notes. 


For  ABODE. 


For  Offsets. 


Sta. 

Bearings. 

Dist. 

1 

N.  20°  E. 

15.50 

2 

E. 

18.00 

3 

S.  20°  E. 

30.00 

4 

W. 

25.00 

5 

N.  32J°W. 

16.09 

Sta. 

Dist. 

Offsets. 

1 

0.00 

2.50 

2 

7.00 

6.00 

3 

12.20 

4.00 

4 

22.25 

7.00 

5 

30.00 

2.55 

Area  =  70  A.  1  R.  33  P. +14  A.  3  R.  8  P.  =  85  A.  1  R.  1  P. 

We    find,    as    in    the    last   article,   ABODE  =  70   A. 
1  R.  33  P. 

To  calculate  the  area  included  between  the  stationary 
line  CD  and  the  line  passing  along  the  middle  of  the 
river,  we   find  Ca  =  7,  ab  ==  Ob  —  Oa  ==  12.20  —  7  = 
5.20,  etc.,  which   gives   the   altitudes  of  the  trapezoids. 
The  parallel  sides  are  given  under  the  head  of  offsets. 

The  altitude  of  a  trapezoid  multiplied  by  the  sum  of 
the  parallel  sides  will  give  twice  its  area. 

The  calculation  is  made  as  in  the  subjoined   table, 
the  letters,  S.,  S.  D.,  0.,  I.  D.,  S.  0.,  D.  T.,  heading  the 


304 


SURVEYING. 


columns  of  the  table,  denoting  stations,  station  dis- 
tances or  distances  from  (7,  offsets,  intercepted  distances, 
sum  of  offsets,  and  double  trapezoids. 


& 

S.  D. 

0. 

/.  D. 

S.O. 

D.  T. 

1 

0.00 

2.50 

2 

7.00 

6.00 

7.00 

8.50 

59.5000 

3 

12.20 

4.00 

5.20 

10.00 

52.0000 

4 

22.25 

7.00 

10.05 

11.00 

110.5500 

5 

30.00 

2.55 

7.75 

9.55 

74.0125 

Area,  14  A.  3  R.  8  P.  2)296.0625 

10)  148.03125 

14.803125 

4 

3.212500 

40 

8.500000 

If  the  offsets  fall  within  the  stationary  line,  the  sum 
of  the  trapezoids  must  be  subtracted. 

In  general,  if  the  lines  are  run  so  as  to  keep  the  field 
on  the  right,  the  sum  of  the  trapezoids  must  be  added 
in  case  of  left-hand  offsets,  and  subtracted  in  case  of 
right-hand  offsets. 

In  case  of  -navigable  rivers,  the  bank  is,  in  general, 
the  boundary — the  first  and  last  offsets  become  0,  and 
the  first  and  last  trapezoids  become  triangles,  but  the 
form  of  the  computation  is  the  same. 


303.   Examples. 

1.  Find  the  area  of  the  lot  of  which  the  following 
are  the  field  notes,  and  make  a  plot  of  the  survey. 


AREA  OF  LAND. 


305 


Rectilinear  Area. 

L.H.  Offsets* 

R.H.  Offsets.** 

®& 

Bearings. 

Dist. 

St.Dist. 

Offsets. 

St.Dist. 

Offsets. 

1 

N.  45°  E. 

10.00 

0.00 

1.00 

0.00 

1.10 

2 

N. 

10.00 

6,50 

4.25 

5.62 

4.00 

3 

N.  45°  E. 

10.00: 

12.50 

2.43 

12.62 

5.27 

4 

E. 

10.00 

17,50 

5.17 

17.07 

1.13 

5* 

S. 

31.21 

26.21 

5.83 

o** 

w. 

17.07 

31.21 

1.25 

7 

N.  45°  W. 

10.00 

55.774715  A.  +  12.17075  A.—  6.10160  A.=  61  A.  3  R.  15  P. 

The  left-hand  offsets  were  made  from  the  fifth  course, 
as  indicated  by  the  single  star;  and  the  right-hand 
offsets  from  the  sixth  course,  as  indicated  by  the 
double  star. 

2.  Find  the  area  of  the  lot  of  which  the  following 
are  the  field  notes,  and  make  a  plot  of  the  survey. 


Rectilinear  Area. 

L.H.  Offsets* 

R.H.  Offsets** 

Sta, 

Bearings. 

Dist. 

St.Dist. 

Offsets. 

St.Dist. 

Offsets. 

1 

N.  30°  E. 

20. 

0.00 

0.00 

0.00 

0.00 

2 

E. 

20. 

6.00 

3.00 

6.00 

4.00 

3* 

S.  30°  E. 

20. 

10.00 

2.00 

14.00 

4.00 

i 

4** 

S.  30°  W. 

20. 

15.00 

3,50 

20.00 

0.00 

5 

W. 

20. 

20.00 

0.00 

6 

N.  30°  W. 

20. 

Ans.  102  A.  1  R.  30  P. 


S.  N.  26. 


306 


SURVEYING. 


304.    Pogue's  Method  of  Finding  the  Area, 

This  method  is  illustrated  by  the  following  example : 


1. 

N. 

20° 

2. 

N. 

43° 

3. 

S. 

70° 

4. 

S. 

40° 

5. 

S. 

65° 

6. 

S. 

42° 

7. 

S, 

8. 

S. 

70° 

9. 

N. 

36}: 

E.,  24.50  ch. 
E.,  22.40  ch. 
E.,  25.50  ch. 
W.,  16.58  ch. 
E.,  25.10  ch. 
W.,  13.50  ch. 
14.20  ch. 
W.,  32.15  ch. 
W.,  34.55  ch. 


Make  a  plot  from  the  field  notes,  draw  meridians 
through  the  most  easterly  and  westerly  stations,  and 
parallels  of  latitude  through  the  most  northerly  and 
southerly,  thus  enclosing  the  whole  figure  in  a  rect- 
angle. 

Find,  from  the  traverse  table,  the  latitudes  and  de- 
partures as  in  diagram. 

To  find  xy,  pass  from  the  most  westerly  station,  round 
the  north,  to  the  most  easterly,  taking  the  sum  of  the 
eastings  minus  the  sum  of  the  westings;  and  to  find 
zw,  pass  from  the  most  easterly  station,  round  the  south, 
to  the  most  westerly,  taking  the  sum  of  the  westings 
minus  the  sum  of  the  eastings,  thus : 

xy  ='8.38  +  15.27  -f-  23.96  —  10.66  +  22.75  ==  59.70 
zw  =  9.03  +  30.21  +  20.44  ==  59.68 

2)119.38 
i  (py  +  zw)  —  the  average  base  =  59.69 

To  find  wx,  pass  from  the  most  southerly  station,  round 
the  west,  to  the  most  northerly,  taking  the  sum  of  the 
northings  minus  the  sum  of  the  southings ;  and  to  find 


AREA  OF  LAND.  307 

2/z,  pass  from  the  most  northerly  station,  round  the  east, 
to  the  most  southerly,  taking  the  sum  of  the  southings 
minus  the  sum  of  the  northings,  thus  : 

WB=  27.86+23.02+  16.38  =  67.26 

yz  =   8.72+12.70+10.60+10.03+14.20+10.99  =  67.24 

2)134.50" 
£  (wx  +  yz)  =  the  average  altitude  =  67.25 

Area  of  rectangle  =  59.69  X  67.25  =  4014.1525. 

From  the  area  of  the  rectangle  we  must  deduct  the 
area  included  between  wxyz  and  abcdefghi,  thus  found. 


8.72 


(W  =  1170X1066  = 

kyml  --=  (8.72  +  12.70)  (22.75  —  10.66)  =--  258.9678 


hnzi  =    9.03+    9.08  +  30.21 
.^3X44X27.86  =  ^ 


1633.9631 

abcdefghi  =  4014.1525  sq.  ch.—  1633.9631  sq.  ch. 

=  2380.1894  sq.  ch.  =  238.02  A. 

For  additional  exercises,  work  the  examples  of  arti- 
cles 301  and  303,  and  compare  the  answers  obtained  by 
the  two  methods. 


308  SURVEYING. 

SUPPLYING  OMISSIONS. 
305.    Case  I. 

When  the  bearing  and  length  of  one  side  are  wanting. 

The  wanting  side  must  be  such  that  its  latitude  and 
departure  will  make  the  work  balance.  Hence,  its  lati- 
tude must  be  the  difference  between  the  sum  of  the 
northings  and  the  sum  of  the  southings  of  the  given 
sides,  and  of  the  same  name  as  the  less ;  and  its-  de- 
parture must  be  the  difference  between  the  sum  of  the 
eastings  and  the  sum  of  the  westings  of  the  given  sides, 
and  of  the  same  name  as  the  less. 

Having  found  the  latitude  and  departure  of  the 
wanting  side,  construct  a  right-angle  triangle  by  draw- 
ing on  the  paper,  to  represent  the  latitude,  a  line,  up 
or  down,  according  as  the  latitude  is  north  or  south ; 
and  at  the  terminus  of  the  line,  draw,  to  represent  the 
departure,  a  horizontal  line,  to  the  right  or  left,  accord- 
ing as  the  departure  is  east  or  west,  and  join  the  ori- 
gin of  the  line  representing  the  latitude  with  the  ter- 
minus of  the  line  representing  the  departure,  and  this 
last  line  will  be  the  hypotenuse  which  will  represent 
the  course  or  length  of  the  line  sought,  and  the  angle 
which  it  makes  with  the  vertical  line  will  be  the 
bearing. 

Denote  the  latitude  by  I,  the  departure 
by  d,  the  course  by  e,  and  the  bearing  by  6, 
then  we  have, 

Having  found  the  bearing  and  distance,  enter  them 
in  the  notes  and  find  the  area. 


SUPPLYING  OMISSIONS. 


309 


306.    Examples. 

Supply  the   omissions   in   the    following    field   notes, 
calculate  the  areas,  and  plot  the  surveys. 


1. 


Sta. 

Bearings. 

Di*t. 

I 

N.  18°  E. 

9.25 

2 

N.  71°  E. 

8.33 

3 

S.  43J°E. 

12.37 

4 

S.  36J°W. 

16.00 

5 

Wanting. 

Want'g. 

N.  43°  W.,  14.18  ch. 
23  A.  3  R.  32  P. 


2. 


Sta. 

.  Bearings. 

Dist. 

1 

N.  24°  W. 

15.50 

2 

N.  31°  E. 

17.07 

3 

E. 

20. 

4 
5 

Wanting. 
S.  56°  W. 

Want'g. 
30.30 

f  S.  12i°E.,  12.13  eh, 
Ans-  \  56  A.  3  R.  0  P. 


307.   Case  II. 

When  the  lengths  of  two  sides  are  wanting. 

Revolve  the  field  so  that  one  of  the  sides  whose 
bearing  only  is  given  shall  become  a  meridian,  and 
find,  by  article  285,  the  bearings  of  all  the  sides  in 
their  new  position. 

The  departure  of  the  side  made  a  meridian  will 
then  be  0,  and  the  difference  of  the  sums  of  the 
columns  of  the  departures  will  be  the  departure,  in 
the  new  position,  of  the  other  side  whose  distance  is 
wanting. 

Knowing  the  bearing  and  departure  of  this  side,  we 
can  find  its  distance  and  latitude.  Then  the  differ- 
ence between  the  sums  of  the  columns  of  latitudes 
will  be  the  length  of  the  side  made  a  meridian. 

Revolve  the  field  to  its  original  position,  calculate 
its  area,  and  make  a  plot  of  it;  or,  if  the  area  only 


310 


SURVEYING. 


is  required  after  supplying  omissions,  it  may  be  com- 
puted more  readily  without  revolving  the  field  to  its 
original,  position. 


308.   Examples. 


1. 


2. 


Sta. 

Bearings. 

Dist. 

1 

N.  30°  E. 

10.00 

2 

N.60°  E. 

18.18 

3 

S.  40°  E. 

Want'g. 

4 

S.  30°  W. 

Want'g. 

5 

W. 

15.00 

6 

N.  18|°W. 

-  19.92 

Sta. 

Bearings. 

Dist. 

1 

N.  47°  E. 

15.65 

2 

S.  57°  E. 

10.55 

3 

S.  28J°W. 

Want'g. 

4 

S.  29J°W. 

1.11 

5 

S.  54°  W. 

1.04 

6 

N.  40J°W. 

Want'g. 

r3d.  20.08  ch. 
Ans.<  4th.  24.52  ch. 
ISO  A.  1  R.  25  P. 


r3d.  17.69  ch. 
Aw.  <  6th.  16.01  ch. 
123  A.  1  R.  14  P. 


309.   Case  III. 

When  the  bearings  of  two  sides  are  loanting. 

If  the  sides  whose  bearings  are  wanting  are  separated 
from  each  other  by  one  or  more  intervening  sides,  sup- 
pose one  of  these  sides  and  a  side  adjacent  to  the  other 
to  change  places,  so  as  to  bring  the  sides  under  con- 
sideration together  without  changing  the  bearings  or 
lengths  of  the  sides  transposed. 

Then,  throwing  these  sides  out  of  consideration, 
find,  by  Case  I,  the  bearing  and  length  of  the  line 
joining  the  extremities  of  the  sides  whose  bearings 
are  wanting. 

This  line  with  those  sides  form  a  triangle  whose  sides 
are  known,  from  which  the  angles  can  be  computed. 

Knowing  the  angles  and  the  bearing  of  one  side, 
the  bearings  of  the  other  sides  can  be  found. 


SUPPLYING   OMISSTOSS. 


311 


Restore  to  their  original  position  the  sides  which 
have  changed  places,  if  such  is  the  fact,  calculate  the 
area,  and  make  a  plot  of  the  field. 


310.    Examples. 


1. 


2. 


Sta. 

Bearings. 

Dist, 

1 

N.  45°  W. 

20.00 

2 

N.  18°  E. 

12.25 

3 

E. 

12.80 

4 

N.  32°  E. 

6.50 

5 

S.  42J°E. 

13.20 

6 

Wanting. 

14.75 

7 

Wanting. 

16.30 

Sta. 

Bearings. 

Dist. 

1 

N.58°  E. 

12.97 

2 

S.  27|°E. 

3.30 

3 

S.  85J°E. 

11.65 

4 

S.  19°  E. 

15.56 

5 

Wanting. 

14.03 

6 

N.  64°  W. 

14.86 

7 

Wanting. 

11.23 

r6th.  S. 

Ans.  <  7th.  S.  65J< 
1 59  A. 


W. 


(  5th.  S.  66J°  W. 

.  <  7th.  N.  15J°  W. 

U5  A.  2  R.  5  P. 


311.   Case  IV. 


Wlien  the  bearing  of  one  side  and  the  length  of  another  are 
wanting. 

Revolve  the  field  so  that  the  side  whose  bearing  only 
is  given  shall  become  a  meridian. 

The  departure  of  this  side  will  then  be  0,  and  the 
difference  of  the  sums  of  the  columns  of  departures 
will  be  the  departure,  in  its  new  position,  of  the  side 
whose  bearing  is  wanting. 

Knowing  the  length  and  departure  of  this  side,  its 
bearing  and  latitude  can  be  found. 

Then  the  difference  of  the  sums  of  the  columns  of  lati- 
tudes will  be  the  length  of  the  side  made  a  meridian. 
.  Revolve   the   field   to   its  original   position,  compute 
the  area  and  plot  the  work. 


312 


SURVEYING. 


Remark  1. — In  finding  the  bearing  of  the  side  whose 
distance  only  is  given,  though  the  angle  can  be  readily 
found,  the  bearing,  and  consequently  the  latitude,  may 
be  either  north  or  south,  since  either  will  comply  with 
the  condition.  The  length  of  the  side  whose  bearing 
only  is  given  will  therefore  be  ambiguous,  and  there 
will  be  two  solutions  to  the  problem.  If  but  one 
solution  is  admissible,  the  omission  should  be  supplied 
by  a  remeasurement ;  and  if  the  lost  bearing  or  dis- 
tance can  not  be  taken  directly,  auxiliary  lines  may 
be  run,  and  the  omissions  supplied  by  Trigonometry. 

2.  From  the  fact  that  two  omissions  can  be  supplied, 
the  surveyor  should  not  deem  it  unimportant  to  find 
all  the  measurements  on  the  ground,  since  thus  he 
can  ascertain  the  correctness  of  his  notes  by  balan- 
cing his  work  —  a  test  not  applicable  when  omissions 
are  supplied. 


312.    Examples. 


1. 


Sta. 

Bearings. 

Dist. 

1 

N.  20°  E. 

12.20 

2 

N.  70°  E. 

15.50 

3 

E. 

18.25 

4 

S.  45°  E. 

20.00 

5 

S. 

20.00 

6 

Wanting. 

20.00 

7 

W. 

Want'g, 

8 

N.  30|  °W. 

36.66 

Sta. 

Bearings. 

Dist. 

1 

S.  34°  E. 

4.56 

2 

S.  66|°W. 

13.84 

3 

N.  12|°  E. 

12.15 

4 

Wanting. 

12.30 

5 

N.  58}  °E. 

9.92 

6 

N.  39J°E. 

5.22 

l-r 
I 

S.  45J°E. 

Want'g. 

8 

S.  52i°W. 

10.76 

r6th.  S.  45°  W. 
Ans.  <  7th.  18.25. 

1 188  A.  3  R.  20  P. 


r  4th.  N.  48J°  W. 
Ans.  <  7th.  18.63. 

132  A.  2  R.  26  P. 


LA  YING   0  UT  LAND.  313 

LAYING   OUT   LAND. 

313.    Laying  out  Squares. 

To  lay  out  a  given  quantity  of  land  in  the  form  of  a  square. 
Let  a  be  the  area  of  the  square,  and  x  one  side. 
Then,  x2  =  a,     .  • .  x  =  \/~a~ 

Reduce  the  given  area  to  square  chains,  extract  the  square 
root,  and  the  result  will  be  the  length  of  one  side. 

With   the  chain   and   transit   lay  out   the  square  on   the 
ground. 

EXAMPLES. 

1.  Lay  out  12  A.  3  R.  20  P.  in  the  form  of  a  square. 

2.  Find   the   side   of   a   square   containing  1  A.,  and 
lay  out  the  square  on  the  ground. 

314.   Laying  out  Rectangles. 

1.  To  lay  out  a  given  quantity  of  land  in  the  form  of  a 
rectangle,  one  side  of  which  is  given. 

Let  a  be  the  area  of  the  rectangle,  6  the  given  side, 
and  x  an  adjacent  side. 

Then,  bx  =  a,     .  • .  x  =  -r-  • 

2.  To  lay  out  a  given  quantity  of  land  in  the  form  of  a 
rectangle  whose  length  is  to  its  breadth  in  a  given  ratio. 

Let  a  denote  the  area  of  the  rectangle,  x  its  length, 
y  its  breadth,  and  m  :  n  the  ratio  of  x  to  ?/. 

1_      lam 
\1T' 
y^  f^-- 
«  m 

S.  N.  27. 


314  SURVEYING. 

3.    To  lay  out  a  given  quantity  of  land  in  the  form  of  a 
rectangle  when  the  sum  of  its  length  and  breadth  is  given. 

Let  a  be   the   area  of  the   rectangle,   x  the   length, 
y  the  breadth,  and  .9  the  sum  of  x  and  y. 


=  a. 


4.  To  lay  out  a  given  quantity  of  land  in  the  form  of  a  rect- 
angle when  the  difference  of  the  length  and  breadth  is  given. 

Let  a  denote  the  area  of  the  rectangle,  x  its  length, 
y  its  breadth,  and  d  the  difference  of  x  and  y. 


315.    Examples. 

1.  The  area  of  a  rectangle  is  3  A.,  one  side  is  4  ch. 
Find  an  adjacent  side  and  lay  out  the  rectangle. 

2.  The  area  of  a  rectangle  is  8  A.;   the  length  is  to 
the  breadth  as   3   is   to  2.     Find  the  sides  and  lay  out 
the  rectangle.  Ans.  10.95  ch.  and  7.30  ch. 

3.  The  area  of  a  rectangle  is  4.8  A. ;   the  sum  of  the 
length  and   breadth  is  14   ch.     Find  the  sides  and  lay 
out  the  rectangle.  Ans.  8  ch.  and  6  ch. 

4.  The  area  of  a  rectangle   is   18  A. ;    the   difference 
of  the  length  and  breadth  is  3  ch.     Find  the  sides  and 
lay  out  the  rectangle.  Ans.  15  ch.  and  12  ch. 

316.   Laying  out  Parallelograms. 

1.   To  lay  out  a  given  quantity  of  land  in  the  form  of  a 
parallelogram  when  the  base  is  given. 


LAYING  OUT  LAND.  315 

Let  a  be  the  area,  b  the  base,  and  x  the  altitude. 

Then  bx  —  a.     .  • .  x  =  -=-  • 

o 

Measure  the  base,  from  any  point  of  which  erect  a 
perpendicular  equal  to  the  calculated  altitude. 

Through  the  extremity  of  the  perpendicular  run  a 
line  parallel  to  the  base,  any  point  of  which  may  be 
taken  for  one  extremity  of  the  upper  base,  which  may 
then  be  measured  off'  on  this  line. 

2.  When  one  side  and  an  adjacent  angle  are  given. 

Let  a  be  the  area,  b  the  given  side,  A  the  given  angle, 
and  x  the  other  side  adjacent  to  this  angle. 

Then  bx  sin  A  =  a,     .  • .  x  =  7- -  - — -  - 

b  sin  A 

3.  When  two  adjacent  sides  are  given. 

Let  a  be  the  area,  b  and  c  the  given  sides,  and  x  their 
included  angle. 

Then  be  sin  x  =  a,     .  • .  sin  x  ^=  -j—  • 

be 

Remark. — If  be  =  a,  then  sin  x  =  1,  x  =  90°,  and  the 
parallelogram  becomes  a  rectangle. 

If  be  <  a,  the  solution  is  impossible. 

317.   Examples. 

1.  The  area  of  a  parallelogram   is   6   A.,  the  base  is 
6  ch.     Find  the  altitude  and  lay  out  the  land. 

2.  The  area  of  a  parallelogram  is  12  A.,  one  side  is 
12  ch.,  and  an  adjacent  angle  is  60°.     Find  the  other 
side  adjacent  to  the  given  angle  and  lay  out  the  land. 

-  '•<- 


316  SVEVEYISG. 

3.  The  area  of  a  parallelogram  is  8  A.,  two  adjacent 
sides  are  8  ch.  and  12  ch.  Find  their  included  angle 
and  lay  out  the  land. 

318.    Laying  out  Triangles. 

1.  To  lay  out  a  given  quantity  of  land  in  the  form  of 
a  triangle  when  the  base  is  given. 

Let  a  denote  the  area,  b  the  base,  and  x  the  altitude. 

Then,  \  bx  =  a,    . ' .  x  =  -y-  • 

Measure  the  base,  at  any  point  of  which  erect  a  per- 
pendicular equal  to  the  calculated  altitude. 

Through  the  extremity  of  this  perpendicular  draw 
a  line  parallel  to  the  base.  This  parallel  will  be  the 
locus  of  the  vertex,  any  point  of  which  may  be  taken 
for  the  vertex. 

2.  When  the  base  is  to  the  altitude  in  a  given  ratio. 

Let  a  denote  the  area,  x  the  base,  y  the  altitude,  and 
m  :  n  the  ratio  of  the  base  to  the  altitude. 


/  2  am 


x  :  y  ::  m  :  n.  J2an 

r\"^r 

3.  When  the  triangle  is  equilateral. 

Let  a  denote  the  area  and  x  one  side. 

Then,  .4330127  *«  =  a,       .  x  = 

4.  When  one  side  and  an  adjacent  angle  are  given. 

Let  a  denote  the  area,  b  the   given  side,  x  the  adja- 
cent side,  and  A  the  included  angle. 

Then,  £  bx  sin  A  =  a.     .  • .  x  =  = — r—.  - 

b  sin  A 


L A  YING  O  UT  LAND,  317 

5.   When  two  sides  arc  given. 

Let  a  denote  the  area,  b  and  c  the   given   sides,  and 
x  their  included  angle. 

2  a 
1  hen,  f  oc  sin  x  =  a,     .  • .  sin  x  =  -—  • 


319.    Examples. 

1.  The  area  of  a  triangle  is  3  A.,  the  .base  is  5  ch. 
Find   the    altitude    and    lay   out    the    triangle    on    the 
ground. 

2.  The  area  of  a  triangle  is  12  A.,  the  base  is  to  the 
altitude  as  3  is  to  2.     Find  the  base  and  altitude  and 
lay  out  the  triangle  on  the  ground. 

3.  The  area  of  an  equilateral  triangle  is  1  A.    Find 
a  side  and  lay  out  the  triangle. 

4.  The  area  of  a  triangle  is  1.2  A.,  one  side  is  2  ch., 
an  adjacent  angle  is  45°.    Find  the  other  side  adjacent 
to  the  given  angle  and  lay  out  the  land. 

5.  The  area  of  a  triangle  is  2  A.,  two  sides  are  6  ch. 
and  10  ch.     Find   the   included  angle  and  lay  out  the 
triangle.  _ 

320.   Laying  out  Circles  or  Regular  Polygons. 

1.  Let  a  be  the  area  of  the  circle,  and  x  the  radius. 
Then,  3.1416  *t  =  a,    .-.  x  = 


2.  Let  a  be  the  area  of  a  regular  polygon,  x  one  side, 
y  one  angle,  n  the  number  of  sides,  and  a'  the  area 
of  a  similar  polygon  whose  side  -is  1.  Article  167. 

[£              180°  (n  —  2) 
Then,  a  x2  —  a,    .*.  x  =  \— ->    y  = -• 


318  SURVEYING. 

321.   Examples. 

1.  Find  the  radius  of  a  circle  whose   area   is   1  A. 
and  lay  out  the  circle. 

2.  Find   the   sides   and   angles  of  a  regular  hexagon 
containing  1  A.  and  lay  out  the  hexagon. 

3.  Find   the   sides   and   angles   of  a   regular  octagon 
containing  1  A.  and  lay  out  the  octagon. 

DIVIDING   LAND. 
322.   Division  of  Rectangles  or  Parallelograms. 

1 .  To  cut  off  a  given  area  by  a  line  parallel  to  a  given  side. 

Let  a  be  the  area,  b  the  given  side,  x  the  distance 
to  be  cut  off  on  the  sides  adjacent  to  ft,  and  A  the 
acute  angle  of  the  parallelogram. 

For  the  rectangle,  bx  =  a,     .' .  x  =  —  • 

For  the  parallelogram,  foe  sin  A  =  a,    ,*.  %  =  j — — — -;• 

2.  When  the  lot  is  to  be  divided  into  parts  having  a  given 
ratio,  by  lines  parallel  to  two  of  the  sides,  divide  the  other 
sides  into  parts  having  the  same  ratio. 

323.   Examples. 

1.  The  sides  of  a  rectangle  are  15  ch.  and  10  ch.;  cut 
off  8  A.  by  a  line  parallel  to  the  shorter  sides. 

2.  The  adjacent   sides  of  a   parallelogram  >are  12  ch. 
and  20  ch.,  and  their  included  angle  is  65°;  cut  off  10 
A.  by  a  line  parallel  to  the  shorter  sides. 

3.  A  man  willed   that    his   farm,  which  was   1    mile 
long   and   J   mile  wide,  be   divided   among   his   three 


DIVIDING   LAND.  319 

sons,  A,  B,  and  C,  aged  21  yrs..  18  yrs..  15  yrs.,  respect- 
ively, in  proportion  to  their  ages,  by  lines  parallel  to 
the  shorter  sides.  Make  the  divisions. 


324.    Division  of  Triangles. 

1.  To  find  a  point  on  a  given  side  of  a  triangle  from 
which  a  line  drawn  to  the  vertex  of  the  opposite  angle  will 
1 divide  the  triangle  into  parts  having  a  given  ratio. 

Let  6  =  AC,  the  given  side; 
D,  the  required  point;   x  =  AD. 

and  ABD   :  DEC  :  :  m  :  n.  A 


By  composition  we  have, 

ABC  :  ABD  :  :  m  -f  n  :  m;  but  ABC  :  ABD  : :  b  :  x. 

bm 


Hence,  m  -\-  n  :  m  :  :  b  :  a-,     .  • .  x  = 


m  -f-  n 


2.   Two    sides   of  a    triangle   being   given,   to    divide    the 
triangle  into  parts  having  a  given  ratio  by  a  line  parallel 

to  the  third  side. 

o 

Let  a  =  BC,  b '==  AC,  the  given  sides;  s 

and  DEC  :  ABED  :  :  m  :  n. 
By  composition  we  have, 

ABC  :  DEC  ::  m+n:  m; 
but  ABC  :  DEC  : :  a2  :  x2  : :  b2  :  y2. 


m  4-  n  :  m  ::  a2  :  x2. 


If,  for  example,  the  triangle  is  to  be  divided  into  three 
equal  parts  by  lines  parallel  to  the  third  side,  then, 


320  SURVEYING. 

The  distances  cut  off  on  a  are  a  ]  ^  a 
The  distances  cut  off  on  b  are  b  i/       b 


3.  Two  cStV/es  o/  a  triangle  being  gire,n,  to  cut  off,  by  a  line 
intersecting  the  given  sides,  an  isosceles  triangle  having  a  given 
ratio  to  the  given  triangle. 

Let  b  =  AC,  c  ~  =  AB,  the  two  given 
sides;    x  =  AE  ==  AD,  and 

ADE  :  ABC  :  :  m  :  n. 
But,  ^.E  :  ABC  ::  x2  :  be. 

Hence,    m  :  n        :  :  x2  :  be,     . 

,  n 

4.  Two  sides  o/  a  triangle  being  given,  to  cut  off  a  triangle 
having  a  given  ratio  to  the  given  triangle  by  a  line  running 
from  a  given  point  in  one  of  the  given  sides  to  the  other 
given  side. 

Let  b  =  AC,  c  —  AB,  the  given 
sides;  D,  the  given  point;  d=AD, 
x  =  AE,  and  AED  :  ABC  ::  m  :  n. 

But,  AED  :  ABC  :  :  dx  :  be. 

Hence,    m  :  n         :  :  dx  :  be,     .'.   x  =  —,  — 

dn 

»5.  The  three  sides  being  given,  to  divide  the  triangle  into 
three  equal  parts  by  lines  running  from  a  g^ven  point  in 
one  of  the  sides. 

Let  a,  b,  c  be  the  sides  of  the 
triangle,  respectively,  opposite 
the  angles  A,  B,  C;  p  r=  AD, 
q  =  CD,  x  =  AE,  and  y  ==  CF. 


, ,- bc 

3  :  1  : :  be  :  px. 
'  I  3  :  1   :  :  ab  :  qv.  J  ab 

•*  I         n t     


.  \ 
Then         " 

.  ) 


DIVIDING  LAND.  321 


If  x,  thus  found,  is  greater 
than  c,  both  lines  will  intersect 
a.  Then  find  'y  as  above. 

Let  x  =  CE. 

Then,  3  :  2  :  :  aft  :  qx,    .  '  .  x  =  ~ 

If  27,  found  above,  is  greater 
than  a,  both  lines  will  intersect 
c.  Then  find  x  as  in  first  case. 


Let  AF  =  y. 

Then,  3  :  2  : :  be  :  py,    .-.  y  =  - 

6p 

6.  To  divide  a,  triangle  into  four  equal  triangles,  join  the 
middle  points  of  the  sides. 

The  lines  ED,  EF,  and  DF  are, 
respectively,  parallel  to  J?(7,  AC, 
and  AB. 

EBF  ==  EDF,  since  each  is  J  the  parallelogram  ED. 
ADE  =  EDF,  since  each  is  J  the  parallelogram  AF. 
CDF  =  EDF,  since  each  is  J  the  parallelogram  CE. 
I^ence,  the  triangles  are  all  equal,  and  each  is  £  ABC. 

7.  The  bearing  of  two  sides  being  given,  to  cut  off  a  tri- 
angle having  a   given   area  by  a   line  of  a  given  bearing 
intersecting  the  sides  whose  bearings  are  given. 

Let  ADE  be  the  triangle  cut  off, 
a  the  area  of  ADE;  x  =  AD  and 
y  =  AE.  The  angles  A,  D,  E  can 
be  determined  from  the  bearings. 


(  \xy  sin  A  =  a.  \ 

'  \  sin  E  :  sin  D  : :  x  :  y.  )  '   '  j 


2  a  sin  E 

x  =• 


rr\-\        j  %  My  "A"  •*•*•  —  **•  ii  •  sin  A  sin  u 

2  a  sin  D 


_         $ 
\"si 


sin  ^4  sin  E 


322 


SURVEYING. 


8.   To   divide   a   triangle  into   two   equal  parts  by   lines 
running  from  a  point  within. 

Let  ABC  be  the  given  triangle, 
and  P  the  given  point. 

-A.'' 

Run  a  line  from  P  to  the  vertex 
A,  and  another  from  P  to  D,  the  middle  point  of  the 
opposite  side  BC.  Run  DE  parallel  to  PA,  and  run  PE. 
PD  and  PE  will  be  the  dividing  lines,  and  CDPE  will 
be  \  ABC. 

For,  draw  the  line  AD,  then  we  have, 

CDPE  =  CDE  +  PED,  and  ACD  =  ODE  +  AED. 
But  PED  =   AED,    .  - .  CDPE  =     ACD. 
.-.  CDPE  = 


9.  Through  a  given  point,  within  a  given  tnangle,  to  draw 
a  line  which  shall  cut  off  a  triangle  having  a  given  ratio  to 
the  given  triangle.  B 

Let  ABC  be  the  given  triangle ; 
a,    6,    c,    the    sides    opposite    the 
angles   A,    B,    C,   respectively;    D 
the  point  given  by  knowing  p  = 
AF  =  ED,   parallel    to    AC;    q  =   A 
AE  =  FD,    parallel    to    AB;    x  =  AH,    y  =  AG,   and 
AGH  :  ABC  ::  m  :  n.     Then, 


-v 

xy  :  be  : :  m  :  n.    j 


bcm  ±  V  b2c2m2  — 4  bcmnpq 


y 


2  bcmq 


bcm  ±  V  b2c2m2  —  4  bcmnpq 


Remark.  —  If  either  x  >  b  or  y  >  c,  the  line  cuts  off 
the  triangle  from  another  angle;  and  the  distances  cut 
off  from  the  vertex  of  this  angle  can  be  found  in  a 
manner  similar  to  the  above. 


DIVIDING   LAND.  323 

10.  To  find  a  point  within  a  triangle  from  ivhich  the 
lines  drawn  to  the  vertices  will  divide  the  triangles  into 
three  equal  triangles. 

Let  ABC  be  the  triangle.  Take 
AD  =  J  AB;  CE  =  J  CB,  and  draw 
DE.  Take  BF^=%  BA,  CG  =  ^CA, 
and  draw  FG. 

P,  the  intersection  of  these  lines, 
will  be  the  point  required. 

For  AD  :  AB  :  :  altitude  of  APC  :  altitude  ABC. 
But  AD  =  \AB,  .-  .  altitude  APC  =  $  altitude  ABC. 

.-.     APC  =  i  ABC. 
In  like  manner,    BPC  =  J  ABC. 


Remark.  —  If  APC,  BPC,  and  APE  are  to  be  to  each 
other  as  p,  q,  r,  take  AD  =  P       ^  of  AB,    CE  = 

—  p  --  i 


of  CB,  BF  =  --  -  —  of  54,  C6?  = 


of  OA,  and  draw  D£"  and  FG,  their  intersection  will  be 
the   point  required. 


Examples. 

1.  One  side  of  a  triangle  is  15  ch.;   from  what  point 
in  this  side  must  a  line  be  drawn  to  the  vertex  of  the 
opposite   angle   so   as   to  divide   the   triangle  into  two 
triangles  which  are  to  each  other  as  2  to  3? 

Ans.  6  ch.  from  one  extremity. 

2.  Two  sides  of  a  triangle   are    10   ch.  and   15  ch., 
respectively;    find  the  distance  from  the  vertex  of  the 


324  SUEVEYJNO. 

angle  included  by  these  sides,  cut  off  on  each  of  these 
sides  by  a  line  parallel  to  the  third  side,  dividing  the 
triangle  into  a  triangle  and  a  trapezoid,  so  that  the 
triangle  cut  off  shall  be  to  the  trapezoid  as  9  to  16. 

Ans.  6  ch.  and  9  ch. 

3.  Two  sides   of   a   triangle   are   4   ch.  and  9  ch.,  re- 
spectively ;    find   the   distance   from  the  vertex  cut  off 
on  each  of  these  sides   by  a  line   cutting  off  an    isos- 
celes triangle  which   shall  be   to  the  given  triangle  as 
16  to  25.  Ans.  4.80  ch. 

4.  Two   sides  of  a  triangle   are   7   ch.  and   9   ch.,  re- 
spectively.    From  a  point  in   one  side,  5  ch.  from  the 
vertex  of  the  angle  included  by  these  sides,  a  line  is 
run   to   the   other    given    side,   cutting   off  a   triangle 
which   is   to   the   given   triangle   as  5   to   9.     How  far 
from  the  same  vertex  does  this  line  intersect  that  side  ? 

Ans.  7  ch. 

5.  The    sides   of    a    triangle,    ABC,   are   a  =-  6    ch., 
b  =  12  ch.,  and  c  =  9  ch.     From   the   middle  point  of 
6  two  lines   are   run,  dividing   the.  triangle  into  three 
equal  parts.     To  what  points  of  what   sides   must   the 
lines  be  run? 

Ans.  To  c,  6  ch.  from  A,  and  to  <7,  4  ch.  from  C. 

6.  The  sides  of  a  triangle,  ABC,  are  a  =  10  ch.,  6  = 
12  ch.,  and  c  =  4  ch.     From   a   point   in  fr,  3  ch.  from 
A,  two  lines  are  run,  dividing  the  triangle  into  three 
equal  parts.     To  what  points  of  what  side  must   these 
lines  be  run? 

Ans.  To  a,  8.89  ch.  from  C,  and  to  o,  4.44  ch.  from  C. 

7.  The  sides  of   a  triangle,  ABC,  are  a  =  5  ch.,  b  = 
18  ch.,  and  c  =  15  ch.    From  a  point  in  6,  12  ch.  from 
A,  two   lines  are  run,  dividing  the  triangle  into  three 
equal  parts.    To  what  points  must  these  lines  be  run? 

Ans.  To  C,  7.50  ch.  from  A,  and  to  B. 


DIVIDING  LAND.  825 

8.  In  the  triangle  ABC,  the  side  AB  runs  N.  50°  E., 
AC  runs  E.     DE,  running   N.  10°  W.,  intersects  these 
lines  in  D  and  E,  and  cuts  off  ADE=  10  A.     Required 
AD  and  AE.  Ans.  AD  =  16.54,  AE  =  18.81. 

9.  In   the   9th    general   problem   of   the    last    article, 
b  =  10  ch.,  c  =  12  ch.,  m  =  1,   n  —  4,  jo  —  2  ch.,   q  == 
3  ch.     Find  a;  and  y.     ^4ns.  a;  ±=  7.24  ch.,  y  ^4.14  ch. 


326.   Division  of  Trapezoids. 

1.   Given  the  bases  and  a  third  side  of  a  trapezoid,  to  divide 
it  into  parts  having  a  given  ratio  by  a  line  parallel  to  the  bases. 

Let  ABCD  be  the  trapezoid,  b=AB,  G 

b'  =  CD,  s  =  AD,  x  ==  AE,  y  =  EF,  the  /\ 

dividing    line,    parallel    to    the    bases,  /    \ 
and  ABFE  :  EFCD  : :  m  :  n. 


Produce  AD  and  BC  to  G. 


/  ABG  :  DCG  ::  b2  :  V*. 

Then,  .  .  .     2  .    ,2 


, 
J        \       \ 


These  proportions  taken  by  division  give, 
ABCD  :  DCG  :  :  b2  —  6'2  :  6'2, 
EFCD   :  DCG  :  :  y2  —  ft'2  :  6/2. 

Since  the  consequents  are  the  same,  we  have, 
ABCD  :  EFCD  :  :  62   -  6'2  :  y2—b'2. 

This  proportion  taken  by  division  gives, 

ABFE  :  EFCD  :  :  b2  —  y*  :  y2  —  b'2, 
But  ABFE  :  EFCD  :  : 


326  SURVEYING. 

Drawing  DH  parallel  to  BC,  we  have, 
AH  :  El : :  AD  :  ED, 

or  b  —  b'  :  y  —  V  ::  s  :  s  —  x,     .'.     x  =    _^_  ,  (6  —  y). 


771  -|-  71 


2.  Given  a  side  and  two  adjacent  angles  of  a  tract  of  land, 
to  cut  off  a  trapezoid  of  a  given  area  by  a  line  parallel  to 
the  given  side. 

1st.  When  the  sum  of  the  two  angles 
<  180°. 

Let  a  =  =  ABCD  =  the  area  cut  off, 
b  =  AB  the  given  side,  x  =  AD,  y  =  BC, 
z  =  DC,  E=  180°  —  (A  +  B). 


(1)     Area  ABE=  J  EB  X  EA  sin  E. 

sin  E  :  sin  A  : :  b  :  EB,    .'.    EB=     f111^- 

sin  E 

sin  E  :  sin  B  : :  b  :  EA,    .  • .    EA  —     Smr,  • 

sin  L 

Substituting  the  values  of  EB  and  EA  in  (1),  we  have, 
b2  sin  A  sin  B 


(2) 


2  sin  E 


.'.    (3)     j>cg  =  r_ 

2  sin  £ 


But    ^45^  :  DCE  : :  62  :  z2. 
62  sin  A  sin  £    62  sin  A  sin 


2  sin  E  2  sin 


sn 


sin  ^4  sin  B 


DIVIDING   LAND.  327 

Draw  DF  parallel  to  EB,  then  ADF=  E  and  DFA  =  B. 

(b  -z)sin£ 
sin  E  :  sin  B  : :  b  —  z  :  x,  . ' .    x  =     — ~ 


(b  —  z)  sin  A 
In  like  manner  we  shall  find      y=         .      ., 

Dill    -/-> 

Since  z  is  known,  x  and  y  are  known. 

2d.  When  the  sum  of  the  two  angles  >  180°. 

E  and  DC  lie  on   opposite   sides  of     ^ 
AB. 

Let  a  ===  .4£CYD  ===  the  area  to  be  cut 
off,  b  =  AB  the  given  side,  x  =  AD,  y  = 
BC,  z  =  DC,  E^A  +  5  —  180°. 

By  a  process  similar  to  that  em- 
ployed in  first  case,  we  find, 

2  a  sin  E 


r  sin  A  sin 
(g_—  6)  sin  B 

(z— - 6)  sin  ^4 
"sin  E 


3.   To  dm'de  a  trapezoid  into  proportional  parts  by  a  line 
joining  the  bases. 

Let  A  BCD  be  the  trapezoid,  6 
and  b'  the  bases,  a  the  altitude, 
m  and  n  the  ratio  of  the  parts. 


p   E 


Take   ,4E= 


>  then  F°= 


328  SURVEYING. 


Then,  AEFD  =  >  ,    and  JJ5CF  =  tt"  (6  +  6'> 

2  (m  -f  n)  2  (m  -j-  w) 

am  (6  +  6')      an  (6  +  6') 


F  c  - 

2  (m  +  n)        2  (m  -\~  n) 
.-.     AEFD  :  EBCF  ::  m  :  w. 

Remark.  —  If  the  line  is  to  be  drawn  from  a  given 
point  P,  in  one  base,  first  divide  as^  above  ;  then,  if  P  is 
on  one  side  of  £",  take  P'  as  far  on  the  other  side  of* 
F,  and  draw  PP'. 

This  change  in  the  dividing  line  does  not  affect  the 
altitude  of  the  parts,  nor  the  sum  of  their  bases,  since 
one  is  increased  as  much  as  the  other  is  diminished, 
nor,  consequently,  their  area. 

A  similar  process  can  be  employed  whatever  be  the 
number  of  parts. 

327.   Examples. 

1.  A  trapezoid  whose  bases   are  b  =  15  ch.  and  b'  = 
12  ch.,  and  third  side  s  —  10  ch.,  is  divided  by  a  line 
parallel  to  the  bases  into  two  parts,  such  that  the  part 
adjacent  to  b  is  to  the   part   adjacent   to  b'  as   3   to   2. 
Required  the  length  of  the  dividing  line,  and  the  dis- 
tance from  b  cut  off  on  s.     Ans.  13.28  ch.,  and  5.73  ch. 

2.  Given  a  side  14.30  ch.,  and  the  two  adjacent  angles, 
60°  and  70°,  respectively,  of  a  tract  of  land  from  which 
10  A.  are  to  be  cut  off  by  a  line  parallel  to  the  given 
side.     Required   the    length   of   the   dividing   line,  and 
the  respective  distances  from  the  given  side  cut  off  on 
the  adjacent  sides. 

Ans.  4.05  ch.,  12.60  ch.,  and  11.61  ch. 

3.  Given  a  side  10  ch.,  and  the  two  adjacent  angles, 
120°   and    115°,  respectively,  of  a   tract  of  land,  from 
which  15  A.  are  to  be  cut  off  by  a  line  parallel  to  the 


DIVIDING  LAND.  329 

given  side.  Required  the  length  of  the  dividing  line, 
and  the  respective  distances  from  the  given  side  cut 
off  on  the  adjacent  sides. 

Ana.  20.32  ch.,  11.42  ch.,  10.91  ch. 
4.  A  trapezoid  whose  parallel  sides  are  AB  =  14  ch., 
and  DC  =-  7  ch.,  is  divided   by  the   line  PP'  into  two 
parts  which  are  to  each  other  as  3  to  4 ;    AP  =  4  ch., 
find  DP'.  Am.  5  ch. 

328.    Division  of  Trapeziums. 

1.  Given  a  side,  two  adjacent  angles,  and  the  area  of  a 
trapezium,  to  divide  it,  by  a  line  parallel  to  the  given  side, 
into  parts  having  a  given  ratio. 

Let  ABCD  be  the  trapezium  ;  b  =  AB, 
the  given  side;  A  and  B,  the  given 
angles;  G  ==  180°  —  (A  +  B),  a  =  the 
area  of  ABCD,  x  =  AE,  y  =  BF,  and 
ABFE  :  EFCD  :  :  m  :  n. 


m  -J-  n  m-\-  n 

ABG  =  %BGX  AGX  sin  G. 

D~       6  sin  yl  ft  sin  7? 

.B(r  =     .     ^    and  AG  =  —. — 77-- 
sin  G  sin  G 

62  sin  A  sin  B 
2  sin  G 

„     b2  sin  A  sin  I?          ma 

iLf  Or    =   — ~ ;;       "^  ~ • 

2  sin  (r  m  -(-  w 

JB6r  :  EFG  ::  AG2  :  'EG2,  ABG  :  EFG  ::  ^G2  :  TG2. 

Substituting,  in  the  proportions,  the  values  of  ABG, 
EFG,  AG  and  BG,  find  EG  and  FG,  and  substituting 
the  values  of  AG,  EG,  BG  and  FG  in  the  equations, 

x  ==  ^G  --  EG  and  y  =  BG  —  FG,  we  have, 
S.  N.  28. 


330  SURVEYING. 


b  sin  B  Ib2  sin2  B  2  ma  sin  B 

/v»      .     ( ^    I  

sin  G  *    sin2  G          (ra  -j-  n)  sin  J.  sin  G 


b  sin  ^4  Ib2  sin2  ^4  _ 

^iVTTT     "  V  sin2  G 


2  ma  sin 


sin  G  *    sin2  G  (m-\-n)  sin  1?  sin  G 


D 


2.  6rwm  the  bearings  of  three  adjacent  sides  of  a  tract 
\of  land,  and  the  length  of  the  middle  side,  to  cut  off,  by  a 
lline  running  a  given  course,  a  trapezium  of  a  given  area. 

Let  a  =  ABCD,  the  area  cut  off;  b  = 
AB,  the  given   side ;    x  =  AD,    y  =  BC, 
z  =  CD. 

From    the    given    bearings,    find    the 
angles  A,  B,  C,  D,  E. 

„_,      b  sin  A        ,    A  „      b  sin  B 
BE  =  —. — ^r  and  AE  =  — — ^-  • 
sin  E  sin  E 

A  _,  _       b2  sin  A  sin  B 

ABE  =  ^BEX  AEXsin  E=  ^—. — = 

z  sin  hi 

b2  sin  A  sin  B 

= s — : — pi a. 

2  sin  E 

z  sin  (7  z  sin  7) 


sin  (7  sin  D        62  sin  ^4  sin  B 


2  sin  E  2  sin  E 

b2  sin  ^4  sin  B          2  a  sin  E 


sin  C  sin  D  sin  C  sin  Z) 


Substituting  the  value  of  z  in  the  values  of  DE  and 
CE,  then  the  values  of  AE,  DE,  BE  and  CE  in  the 
equations, 


DIVIDING  LAND.  331 

x  =  A  E  —  DE,  and  y  =  BE  —  CE,  we  find, 

b  sin  B  /62  sin  ^4  sin  .#  sin  0          2  a  sin  C~ 


x  = 


-V- 


sin  E          \         sin2!?  sin  D  sin  .D  sin 

62  sin  .4  sin  B  sin  Z)         2  a  sin 


6  sin  .4  / 

E"    '  "  \ 


sin2  E  sin  0  sin  C  sinE 

Remark.  —  If  .4-f-#  >  180°,  the  values  of  x  and  y  are 


sin  yl  sin  B  sin  C          2  ft  sin  C          b  sin 


sin2  E  sin  D  sin  D  sin  E1          sin  E 


V62  sin  A  sin  B  sin  D ' 
sin2  E  sin  C                si 


2  a  sin  D          b  sin 


sin2  E  sin  C  sin  C  sin  E1          sin  £" 

3.  The  bearings  of  several  adjacent  sides  of  a  tract  of  land 
being  given,  and  the  length  of  each,  except  the  first  and  last, 
to  cut  off  a  given  area  by  a  line  of  given  bearing  intersecting 
the  first  and  last  sides. 

Let  the  bearings  and  distances  of 
AK,  KL,  LM,  MN,  NB  be  given,  and 
the  bearings  of  AD  and  EC;  and  let 
a  be  the  area  cut  off' by  CD. 

Draw  ABj  then,  in  the  polygon, 
ABNMLK,  -the  bearings  and  dis- 
tances of  all  the  sides  are  known, 
except  AB,  which  can  be  computed,  and  the  area  of 
ABNMLK  found.  Subtract  the  area  thus  found  from 
the  area  to  be  cut  off  by  CD,  and  the  remainder  will 
be  the  area  of  A  BCD. 

Then,  by  the  last  case,  find  AD  and  BC. 

4.  The  bearings  of  the  sides  of  any  quadrilateral  tract  of 
land  and  the  distances  of  two  opposite  sides  being  given,  to 
divide  it  into  parts  having  a  given  ratio  by  a  line  of  a  given 
course  intersecting  the  other  sides. 


332  SURVEYING. 

Let  b  =  AB,  c  =  CD, 
x  =  .4F,  ?/  ==  BF,  z  ==  FF, 
and.  .4£FF  :  EFCD  ::  m  :  w. 

Find  the  angles  A,  B,  C,  D,  E,  F,  G. 

P  ~       b  sin  A        .„       b  sin  1 
sin  G   '  sin  G 

_  e  sin  I>  _  2  sin  F       ^       z  sin 

C-  (JT  —  .       77~  >       f  (jr  —  .       7~f  ~  j 


.  ,      _  .  , 

Sill  Lr  Sill  LT  Sin  Cr 

sin  ^4  sin  B  —  c2  sin  (7  sin  />) 


sin   T 

sin  .4  sin  B  —  z2  sin  E  sin 

—  ~  —  :  -  7i  — 

2  sin  G 

Equating  these  values  of  ABFE,  we  find, 


??&2  sin  J  sin  B  -f  me2  sin  C  sin  D 

(m  -f  «)  sin  F  sin  F 
Substituting  this  value  of  z  in  the  values  of  FG  and 
FG;  then  the  values  of  ^4G,  FG,  BG  and  FG  in 

x  =  AG  —  EG,  and  y  =  BG  —  FG,  we  have, 

_  6  sin  B       sin  F    Inb2  sin  A  sin  B  -j-  me2  sin  C  sin  /) 
sin  G        sin  G  ^  (m  -+-  «)  sin  F  sin  F 

_  &  sin  A       sin  F    /  n62  sin  A  sin  #  +  me2  sinC  sin  /) 
sin  G        sinG^/  (m  +  n)  sin  F  sinF 

5.  !T^6  bearings  and  distances  of  the  sides  of  any  quad- 
rilateral tract  of  land  being  given,  to  divide  it  into  parts 
having  a  given  ratio  by  a  line  dividing  two  opposite  sides 
proportionally. 

b  «  AB,   c  --=  CD,   d  ==--  AD, 
e  =BC,    x  =  AE,    y  =  BF, 

ABFE  :  EFCD  ::  m  :  n, 

ex 
x  :  d  —  x  ::  y  :  e  —  y,     .• .  y  =  —  • 


DIVIDING  LAND. 


333 


From  the  bearings  find  the  angles  A,  B,  C,  Z),  G. 

nri       b  sin  A  ,    4ri       b  sin  B 

BG  =  — : — 77-  =  p.  and  AG  =  — .     „    =  q. 
sin  G  sin  (r 

m  (b2  sin  A  sin  B  —  c2  sin  C  sin  D) 
2  (m  -f-  n)  sin  (r 

62sin  ^4  sin  7?      m(b2  sin  ^4  sin  B  —  c2  sinCsinD) 
2  sin  G  2  (w  -f-  ri)  sin  G 

nb2  sin  A  sin  B  -f-  ??ic2  sin  (7  sin  D 
2  (m  +  ri)  sin  Gr 

But  EFG  =  %(q  —  x)(p-  --T-)  sin  G. 

«c.     .     „ 
-  x)  (  p T-)  sin  G  =  s. 


Sdes 
sin  G 


dp  -f  eg  ±: 


Sdes 
smG 


6.  y^e  bearings  and  distances  of  the  sides  of  a  quadrilat- 
eral being  given,  to  cut  off  a  given  area  by  a  line  running 
through  a  point  whose  bearing  and  distance  from  the  vertex 
of  one  of  the  angles  are  given. 

Let  a  be  the  area  of  ABFE,  cut  off 
by  EF  through  P. 

b  =AB,    c  =  CD,    u  =  EG, 
v=FG,    x=AE,    y  =  BF. 

The  bearings  give  the  angles  A,  JB, 
C,  D,  PCQ,  PCD. 

n~      b  sin  A        .  „      b  sin  B 
BG——. — ~-,    AG  =  — -. — 77- 
sin  Cr  sin  Or 

_  62  sin  A  sin  B 
2  sinG 


b2  sin  A  sin  B 

— — - — ; — 

2  sin  G 


334 


SURVEYING. 


In  the  triangle  DCP  we  have  given  f7),  OP,  and 
DCP;  hence  CDP  and  DP  can  be  found  ;  then  PDR  = 
CDR  —  CDP. 

PR  =  DP  sin  PDR  =  p,  and  PQ  =  CP  sin  PCQ  =  q. 
EPG  =  \pu,  and  FPG  =  % 


But 


sn 


X  = 


sin  G 
b  sin  A 
sin  G 


7.  The  bearings  and  distances  of  the  sides  of  a  quadrilat- 
eral being  given,  to  divide  it  into  four  equal  parts  by  two 
lines  intersecting  the  pairs  of  opposite  sides,  respectively,  one 
line  being  parallel  to  one  side. 

Let  EF,  parallel  to  AB,  and 
MN,  parallel  to  BC,  each  divide 
A  BCD  into  two  equal  parts ; 
and  PQ,  parallel  to  FC,  divide 
EFCD  into  two  equal  parts. 

Find  AE,  BF,  BM,  CN,  CP,  and  FQ,  by  problem  1  of 
this  article. 

EF  =  AB  —  AE  cos  A  —  BF  cos  B. 

Likewise  find  MN  and  PQ.     NP  =  CN—CP. 

Produce  MQ  to  /,  draw  NH  parallel  to  IM,  and  draw 
HI;  then  will  EF  and  HI  be  the  lines  required. 

The  line  EF  is  evidently  one  of  the  required  lines. 
We  are  now  to  prove  that  HI  is  the  other. 


-.«- 


DIVIDING  LAND.  335 

The  two  triangles,  HNI  and  HNM,  are  equal,  since 
they  have  a  common  base,  HN,  and  a  common  altitude, 
their  vertices  being  in  IM,  parallel  to  the  base. 

To  each  of  these  equal  triangles  add  AHND,  and  we 
have  ARID  =  AMND  =  %ABCD. 

We  are  now  to  prove  that  HI  divides  EFCD,  and  also 
ABFE  into  two  equal  parts. 

IMH  :  IQL  ::  IM2    :  ~IQ\ 

IMN  :  IQP  ::IM2  :  7Q2. 
.  •  .  IMH  :  IQL  :  :  IMN  :  IQP. 
But  IMH  =  IMN.  .  •  .  IQL  =  IQP. 

To  each  add  QFCI,  and  we  shall  have, 
LFCI  =  QFCP  ==  J  EFCD. 
Again,  HBCI=AHID  and  LFCI  =  ELID. 

Subtracting  the  second  from  the  first,  member  from 
member,  we  have, 

HBFL  =  AHLE. 

Hence,  HI  is  the  other  division  line  required. 

Let  us  now  find  the  situation  of  the  points  H  and  /, 
on  the  lines  AB  and  CD,  respectively. 

NM  :  PQ  ::  NP+PI  :  PI. 


(NM—  PQ)  PI=PQX  NP. 

PI  —  ^Q  X  NP      Tnen    nj  —  np  _  pj 
~ 


NM—PQ 

The  bearing  and  length  of  JAf,  and  the  area  of  ICBM, 
can  be  found  by  Art.  305.     IMH  ==  ICBH  —  ICBM. 

If  p  be  the  perpendicular  from  /to  AB, 

MH  .       BH  =  BM  +  ME. 


336  SURVEYING. 

329.   Examples. 

1.  A  trapezium,  one  side  of  which  is  20  ch.,  the  ad- 
jacent angles  60°  and   80°,  respectively,  and   the    area 
10  A.,  is  divided  into  two  'equal  parts  by  a  line  paral- 
lel to  the  given  side.     Required  the  distance  from  the 
given  side  cut  oft'  on  the  adjacent  sides. 

Ans.  3.04  ch.,  and  2.08  ch. 

2.  From   a    tract  of  land,   the   bearings  of  three  of 
whose  adjacent  sides  are  S.  20°  W.,  E,  and  N.  10°  W., 
and  the  distance  of  the  middle  side  is  10  ch.,  5  A.  are 
cut  off  by  a  line  running   S.  70°  W,  and   intersecting 
the  first  and  third  of  the  above  mentioned  sides.     Re- 
quired  the   distances   cut   off  on   these   sides  from  the 
middle  side.  Am.  4.91  ch.,  and  7.29  ch. 

3.  From  a  tract  of  land,  the  bearings  of  whose  sides 
are  S.  38°  E.,   S.  29f°  E.,   S.  31f°  W.,  N.  61°  W.,  and 
N.  10°  W.,  respectively,  and  the  distance  of  the  second, 
third,  and  fourth  sides  are  8.18  ch.,  15.36  ch.,  and  14.48 
ch.,  respectively,  39  A.  2  R,  36  P.,  are  cut  off  by  a  line 
running  N.  80°  E.,  and  intersecting  the  first  and  last 
sides.     Required   the   distances   cut   off  on    these   sides 
respectively.  Ans.  7.01  ch.,  16.19  ch. 

4.  A  tract  of  land,  the  bearing  and  distances  of  whose 
sides   are  AB,  E.  22.21  ch.;    BC,  N. ;    CD,  N.  56J°  W., 
12.  ch. ;  DA,  S.  24°  W.,  is  cut  by  EF  running  S.  76J°  E., 
intersecting  AD  and  BC,  and  dividing  the  field  so  that 
ABFE  :  EFCD  : :  5  :  3.     Required  AE  and  BF. 

Ans.  AE=-~  16.50.  ch.,  BF  =  11.34  ch. 

5.  A    trapezium    whose    sides    are    AB  --  20.45   ch., 
BC  ==  21.73  ch.,   CD  =  13.98  ch.,  DA  a*  13.32  ch.,  and 
whose  angles  are  A  *=  97|°,   B  ==  64°,   C  =  89J°,   D  = 
109°,  is  divided  into  two  equal  parts  by  the   line  EF, 


DIVIDING  LASD.  337 

dividing   AD   and   BC    proportionally.      Required    AE 
and  BF.  Ans.  AE  =  6.22  ch.,  BF  =  10.15  ch. 

6.  Within  a  tract   of   land  whose   sides   are  —  1st.  E. 
45.58  ch.;   2d.  X.  134°  W.,  40.86  ch.;  3d.  S.  82°  W.,  30.40 
ch.,    4th.  S.  9{>°  W.,  36  ch.  —  there   is   a  spring  whose 
bearing  and  distance  from  the  3d  corner  is  S.  21°  W., 
15.80  ch.     It  is  required  to  cut  off  40  A.  from  the  north 
side  of  this  tract  by  a  line  running  through  the  spring 
and   intersecting   the  2d  and  4th  sides.     Required  the 
distance  from  the  1st  corner  to  the  point  of  intersection 
on  the  4th  side.  Ans.  26.73  ch. 

7.  A  tract  of  land  whose  boundaries  are — 1st.  E.  23.24 
ch.;  2d.  N.  11}°  W.,  15.25  ch.;  3d.  N.  5H°  W.,  11.50  ch.; 
4th.  S.  27°  W.,  24.82  ch.— is  to  be  divided  into  four  equal 
parts  by  two  lines,  one  parallel  to  the  first  side,  the  other 
intersecting  the  first  and  third  sides.     Required  the  dis- 
tances cut  off  by  the  parallel  from  the  first  and  second 
corners,  measured  on  the  fourth  and  second  sides,  respect- 
ively;   also  the  distances  cut  off  by  the  other  line  from 
the  first  and  fourth  corners,  measured  on  the  first  and 
third  sides,  respectively. 

Ans.  8.57  ch.,  7.79  ch.,  10.66  ch.,  3.15  ch. 


330.    Division  of  Polygons. 

1.  From  a  given  point  in  the  boundary  of  a  tract  of  land, 
the  bearings  and  distances  of  whose  sides  are  given,  to  run  a 
line  which  shall  cut  off  a  given  area.  c 

Let  A  be  the  point,  and  suppose  it 
probable  that  the  dividing  line  will 
terminate  on  DE.  Suppose  the  closing 
line  AD  to  be  run,  the  bearing  and 
distance  of  which  can  be  found  on  the 
S.  N.  29. 


338  SURVEYING. 

ground  by  observation  and  measurement,  or,  as  in  sup- 
plying omissions,  from  the  bearings  and  distances  of  AB, 
BC,  and  CD.  Compu  e  the  area  of  ABCD,  which,  if 
less  than  the  -area  to  be  cut  off,  subtract  from  that  area, 
which  gives  the  addition,  a,  to  ABCD.  The  bearings 
of  AD  and  DE  give  the  angle  ADE. 

The  perpendicular,  AG=AD  sin  ADG. 

Then,  if  AP  is  the  dividing  line,  DP  t=±  J^TTT  - 


If  DP  >  DE,  run  another  closing  line  AE,  and  pro- 
ceed as  before. 

If  ABCD  is  greater  than  the  area  to  be  cut  off,  sub- 
tract the  area  to  be  cut  off  from  ABCD  and  divide  the 
difference  by  one-half  the  perpendicular  from  A  to  CD, 
and  the  quotient,  if  less  than  DC,  will  be  the  distance 
from  D  to  the  point  on  DC  to  which  the  division  line 
is  to  be  drawn. 

If  the  quotient  is  greater  than  DC,  run  another  clos- 
ing line,  AC,  and  proceed  as  before. 

2.  Through  a  given  point  within  a  tract  of  land,  the  bear- 
ings and  distances  of  whose  sides  are  given,  to  run  a  line  which 
shall  cut  off  a  given  area. 

Let  P  be  the  given  point.  Run  a 
trial  line,  AB,  and  calculate  the  area 
which  it  cuts  off. 

Let   d  -be    the    difference   between 
this   area,  which   we  will    suppose   too   small,  and   the 
area  to  be  cut  off. 

Let  CD  be  the  division  line  required. 


DIVIDING  LAND.  339 

Let  m  =  AP,  and  n  =  PB,  which  measure;   find  the 
angle  PAG,  also  PBD.     We  are  to  find  the  angle  P. 

€--=-  180°  —  (A  +  P)  and  D  ==  180°  —  (B  +  P). 
.  •  .     sin  C-=  sin  (J.  -f  P)  and  sin  Z)  s=  sin  (5  -f-  P). 

p  m  sin  ^4  ??i  sin  P 

"  '  *  sin  (?+  P)  ' 


.p         m2  sin  ^i  sin  P 

: 


.  • .     BPD  = 

/r, 

d  = 


sin(£-f-P)' 
2  sin  B  sin  P 


2  sin 
m2  sin  A  sin  P        n2  sin  B  sin  P 


2  sin  (A  -f-P)  2  sin  (P  -f-P) 


cot  P  +  cot  A        cot  P  -f  cot  B 
Use  natural  co-tangents,  find  cot  P,  and  then  P. 

331.    Examples. 

1.  The  boundaries  of  a  tract  of  land  are :   AB,  W.  25 
ch. ;   PC,  N.  32i°  W.,  16.09  ch. ;  CD,  N.  20°  E.,  15.50  ch. ; 
DE,  E.  25  ch. ;    EF,  S.  30°  E. ;   and  FA,  S.  25°  W.,  to 
the  point  of  beginning.    A  line  is  run  from  A,  cutting- 
off  70  A.  1  R.  33  P.  from  the  west  side.     Required  the 
second  point  in  which  this  line  cuts  the  boundary. 

An?.  The  side  'DE,  18  ch.  East  of  D. 

2.  It  is  required   to   run   a   line   through  a  point,  P, 
within   a   field,   so  as   to   cut   off  10  A.     A  guess  line 
through  P,  intersecting  opposite  sides  in  A  and  J5,  cuts 
off  9  A.     Required   the   angle  which  the  true  division 
line,  CD,  makes  with  AB,  if  AP  =  12  ch.,  PB  =  4  ch., 
PAC  —  90°,  PBD  =  60°.  Ans.  8°  48'. 


340  SURVEYING. 

LEVELING. 
332.   The  Y  Level. 

The  Y  level,  so  called  from  the  form  of  the  support? 
in  which   the   telescope   rests,  is   exhibited   in  the  ar, 
nexed  engraving. 

The  telescope  is  inclosed  in  rings,  by  which  it  can  be 
revolved  in  the  Y's  or  clamped  in  any  position. 

The  Y's  have  each  two  nuts,  adjustable  with  the  steel 
pin,  and  the  rings  are  clamped  in  the  Y's  by  bringing 
the  clips  firmly  on  them  by  means  of  tapering  Y  pins. 

The  interior  construction  of  the  telescope  is  exhibited 
in  the  following  figure. 


The  rack  and  pinion,  A  A  and  C(7,  are  contrivances, 
the  first  for  centering  the  eye-piece,  and  the  second  for 
insuring  the  accurate  projection  of  the  object-glass  in 
a  straight  line. 

The  level  is  a  ground  bubble  tube,  attached  to  the 
under  side  of  the  telescope,  and  furnished  at  each  end 
with  arrangements  for  the  usual  movements  in  both 
horizontal  and  vertical  directions. 

The  tripod  head  is  similar  to  that  in  the  transit. 

333.    Adjustments. 

1.  To  adjust  the  line  of  collimation,  set  the  tripod 
firmly,  remove  the  Y  pins  from  the  clips,  so  that  the 
telescope  shall  turn  freely,  clamp  the  instrument  to 


(341) 


342  SURVEYING. 

the  tripod  head,  and  by  means  of  the  leveling  and 
tangent  screws,  bring  either  of  the  wires  to  bear  on 
a  clearly  marked  edge  of  an  object,  distant  from  two 
to  five  hundred  feet. 

Turn  the  telescope  half-way  round,  so  that  the  same 
wire  is  brought  to  bear  on  the  same  object. 

Should  the  wire  not  range  with  the  object,  bring  it 
half-way  back  by  moving  the  capstan  head  screws,  BB, 
at  right  angles  to  it,  in  the  opposite  direction,  on 
account  of  the  inverting  property  of  the  eye-piece,  and 
repeat  the  operation  till  it  will  reverse  correctly. 

Proceed  in  like  manner  with  the  other  wire. 

Should  both  wires  be  much  out,  adjust  the  second 
cifter  having  nearly  completed  the  adjustment  of  the 
first,  then  complete  the  adjustment  of  the  first.  • 

To  bring  the  intersection  of  the  wires  into  the  center 
of  the  field  of  view,  slip  off  the  covering  of  the  eye- 
piece centering  screws,  shown  at  AA,  and  move,  with 
a  small  screw-driver,  each  pair  in  succession,  with  a 
direct  motion,  as  the  inversion  of  the  eye-piece  does 
not  affect  this  operation,  till  the  wires  are  brought,  as 
nearly  as  can  be  judged,  into  the  required  position. 

Test  the  correctness  of  the  centering  by  revolving 
the  telescope  and  observing  whether  it  appears  to  shift 
the  position  of  an  object. 

If  the  position  of  the  object  is  shifted  by  revolving  the 
telescope,  the  centering  is  not  perfectly  accomplished. 

Continue  the  operation  till  the  centering  is  perfect. 

2.  To  adjust  the  level  bubble,  clamp  the  instrument 
over  either  pair  of  leveling  screws,  and  bring  the 
bubble  to  the  middle. 

Revolve  the  telescope  in  the  Y's  so  as  to  bring  the 
level  tube  on  either  side  of  the  center  of  the  level  bar. 


LEVELING.  843 

Should  the  bubble  run  to  one  end,  rectify  the  error 
by  bringing  it,  as  nearly  as  can  be  estimated,  half-way 
back  with  the  capstan  screws  in  the  level  holder. 

Again  bring  the  level  over  the  center  of  the  bar, 
and  bring  the  bubble  to  the  center;  turn  the  level  to 
one  side,  and,  if  necessary,  repeat  the  operation  till  the 
bubble  will  keep  its  position  when  the  tube  is  turned 
to  either  side  of  the  center  of  the  bar. 

Now  bring  the  bubble  to  the  center  with  the  level- 
ing screws,  and  reverse  the  telescope  in  the  Y's  with- 
out jarring  the  instrument.  Should  the  bubble  run  to 
either  end,  lower  that  end,  or  raise  the  other  by  turn- 
ing small  adjusting  nuts  at  one  end  of  the  level  till, 
by  estimation,  half  the  correction  is  made. 

Again  bring  the  bubble  to  the  middle,  and  repeat  the 
operation  till  the  reversion  can  be  made  without  caus- 
ing any  change  in  the  bubble. 

3.  To  adjust  the  Y's,  or  to  bring  the  level  into  a  posi- 
tion at  right  angles  with  the  vertical  axis,  so  that  the 
bubble  will  remain  in  the  center  during  an  entire 
revolution  of  the  instrument,  bring  the  level  tube 
directly  over  the  center  of  the  bar,  and  clamp  the  tele- 
scope in  the  Y's,  placing  it,  as  before,  over  two  of  the 
leveling  screws,  unclamp  the  socket,  level  the  bubble, 
and  turn  the  instrument  half-way  around,  so  that  the 
level  bar  may  occupy  the  same  position  with  respect  to 
the  leveling  screws  beneath. 

Should  the  bubble  run  to  either  end,  bring  it  half- 
way back  by  the  Y  nuts  on  either  end  of  the  bar. 

Now  move  the  telescope  over  the  other  set  of  level- 
ing screws,  bring  the  bubble  again  into  the  center,  and 
proceed  as  before,  changing  to  each  pair  of  screws,  suc- 
cessively, till  the  adjustment  is  nearly  completed,  which 
may  now  be  done  over  a  single  pair  of  screws. 


344  SURVEYING 

334.   The  Use  of  the  Level. 

Set  the  legs  firmly  in  the  ground,  test  the  adjust- 
ments, making  corrections  if  necessary. 

Bring  the  wires  precisely  in  the  focus,  and  the  object 
distinctly  in  view,  so  that  the  spider  lines  will  appear 
fastened  to  the  surface  of  the  object,  and  will  not  change 
in  position  however  the  eye  be  moved. 

The  bubble  resting  in  the  middle,  the  intersection  of 
the  spider  lines  will  indicate  the  line  of  apparent  level. 

335.    Leveling  Rod. 

The  New  York  Leveling  Rod,  represented 
in  the  engraving  with  a  piece  cut  out  of  the 
middle,  so  that  both  ends  may  be  exhibited, 
consists  of  two  pieces,  one  sliding  from  the 
other. 

The  graduation  commences  at  the  lower  end, 
which  is  to  rest  on  the  ground,  and  is  made 
to  tenths  and  hundredths  of  a  foot. 

A  circular  target,  divided  into  quadrants 
of  different  colors,  so  as  to  be  easily  seen, 
moves  on  the  front  surface  of  the  rod,  which 
reads  to  six  and  one-half  feet. 

If  a  greater  height  is  required,  the  horizon- 
tal line  of  the  target  is  fixed  at  6J  feet,  on 
the  front  surface,  and  the  upper  part  of  the 
rod,  which  carries  the  target,  is  run  out  of 
the  lower,  and  the  reading  is  obtained  on  the 
graduated  side  up  to  an  elevation  of  twelve  ft. 

A  clamp  screw  on  the  back  is  used  to  fasten 
the  rods  together  in  any  position. 


LEVELING.  345 

336.   Definitions. 

A  level  surface  is  the  surface  of  still  water,  or  any 
surface  parallel  to  that  of  still  water. 

Such  a  surface  is  convex,  and  conforms  to  the  sphe- 
roidal form  of  the  earth. 

A  level  line  is  a  line  in  a  level  surface. 

The  difference  of  level  of  two  places  is  the  distance 
of  one  above  or  below  the  level  surface  passing  through 
the  other.  . 

Leveling  is  the  art  of  ascertaining  the  difference  of 
level  of  two  places. 

The  apparent  level  of  any  place  is  the  horizontal  plane 
tangent  to  the  level  surface  at  that  place. 

The  line  of  apparent  level  of  any  place  is  a  horizontal 
line,  tangent  to  a  level  line  at  that  place. 

The  Y  Level  indicates  the  line  of  apparent  level 
and  not  the  true  level,  which  is  a  curved  line. 

The  correction  for  curvature  is  the  amount  of  devia- 
tion for  a  given  distance  of  the  line  of  apparent  level 
from  the  line  of  true  level  to  which  it  is  tangent  at 
the  point  from  which  the  distance  is  measured. 

337.    Problem. 

To  compute  the  correction  for  curvature. 

Let  t  denote  the  tangent,  c  the  cor- 
rection for  curvature,  d  the  diameter 
of  the  earth. 

Then,  by  Geometry,  we  have, 

(<*  +  «)*  =  «',     .-.<  =  —  • 


346  SURVEYING. 

Since   c   is   very   small   compared  with   d,  it   can   be 
dropped  from    the   denominator  without  sensibly  affect- 

ing the  result. 

t* 


The  arc,  which  is  the  distance  measured,  will  not 
differ  perceptibly  from  the  tangent,  for  all  distances  at 
which  observations  are,  made,  and  may  be  substituted 
for  it. 

Calling  another  distance.  ^,  and  the  corresponding 
correction,  c',  we  have, 

/'2 

J=  —•  .'.     c  :  c'  ::  t2  :  tf*. 

a 

1.  The  correction  for  curvature,  for  a  given  distance,  is 
equal   to  the  square  of  the  distance  divided  by  the  diameter 
of  the  earth. 

2.  The  corrections  for  different  distances  are  to  each  other 
as  the  squares  of  the  distances. 

Let  us  find  the  correction  for  the  distance  100  chains, 
calling  the  diameter  of  the  earth  7920  miles. 

1002X66X  12 

7920X80"     ^12-5l»ches- 

The  correction  for  any  other  distance,  for  example,  5 
ch.,  can  be  found  from  the  proportion. 

1002  :  52     ::  12.5  :  c,     .-.  c  =  .031  inches. 
For  1  mile,    1002   :  SO2   :  :  12.5  :  c,     .'.  c=  8  inches. 
For  m  miles,      I2  :  m2    ::$:<?,          .  '.  c  =  8  m2  in. 

A  correction  for  refraction  is  sometimes  made  by  di- 
minishing the  correction  for  curvature  by  J  of  itself. 

If  the  leveling  instrument  is  placed  midway  between 
the  two  places  whose  difference  of  level  is  to  be  found, 
the  curvature  and  refraction  on  the  two  sides  of  the 


LEVELING. 


347 


instrument    balance,    and    the    difference    of    apparent 
level  will  be  the  difference  of  true  level. 


338.    Problem. 

To  find  the  difference  of  level  of  two  places  visible  from  a 
point  midway  between  them  or  from  each  other,  when  the 
difference  of  level  does  not  exceed  twelve  feet. 


Let  A  and  B  be  the  two  places,  and  C  the  place  mid- 
way from  which  both  are  visible. 

Place  the  level  at  (7,  and  let  the  rod-man  set  up  the 
leveling  rod  at  A,  and  slide  the  vane  till  he  learns,  by 
signals  from  the  surveyor  at  the  level,  that  its  hori- 
zontal line  is  in  the  line  of  apparent  level.  Let  the 
height  be  accurately  observed  and  noted,  and  the  rod 
be  transferred  to  B,  and  the  height  observed,  and  noted 
as  before. 

The  difference  of  these  heights  will  be  the  difference 
of  level. 

If  a  gully  intervene,  so  that  the  line  of  apparent 
level,  from  the  intermediate  station,  would  not  cut  the 
rod,  place  the  instrument  at  one  station,  and  take  the 
height  on  the  staff  at  the  other  station  marked  by  the 
vane  when  in  the  line  of  apparent  level,  from  which 
subtract  the  height  of  the  instrument,  and  the  differ- 
ence corrected  for  curvature  and  refraction  will  be  the 
difference  of  level  required. 


348 


SURVEYING. 


339.    Problem. 

To  find  the  difference  of  level  of  two  places  which  differ  con- 
siderably in  level,  or  which  can  not  be  seen  from  each  other. 


Let  A  and  D  be  the  places  whose  difference  of  level 
is  required. 

Place  the  level  at  the  station  L,  midway  between 
two  convenient  points,  A  and  B.  Take  the  backsight 
to  J,  and  note  the  height  of  E.  Send  the  rod  to  #,  and 
note  the  height  of  the  foresight  at  F.  Remove  the 
level  to  M,  note  the  height  of  the  backsight  at  G  and 
the  foresight  at  H.  Remove  the  level  to  A7,  note  the 
height  of  the  backsight  at  /,  and  the  foresight  at  J. 

Then  will  the  difference  of  the  sum  of  the  backsights 
and  the  sum  of  the  foresights  be  the  difference  of  level 
of  A  and  D. 

For,  we  find  for  the  sum  of  the  backsights, 
AE  +  BO  +  CT   ^  AE  +  BF  +  FG  +  CI. 

And,  we  find  for  the  sum  of  the  foresights, 
BF  +  OH  +  DJ=BF  +  01+  IH  +  DJ 
=  BF+CI+PG. 

The  sum  of  the  backsights,  minus  the  sum  of  the  fore- 
sights, =  AE  +  FG  —  PG  =  -AK  =  difference  of  level, 
which  in  the  field  notes  is  denoted  by  D.  L. 


LEVELING. 


349 


If  the  sum  of  the  foresights  exceeds  the  sum  of  the 
backsights,  the  point  D  is  below  A;  if  the  reverse  were 
true,  the  point  D  would  be  above  A,  as  indicated  by 
the  sign. 

It  is  not  essential  that  the  intermediate  stations  be 
directly  between  the  places. 


340.    Field  Notes. 


Stations. 

Backsights. 

Foresights. 

1 
2 
3 

5.40 
3.12 
2.40 

1.50 
5.25 
8.16 

Sums  .  .. 
D.  L.  = 

10.92 
14.91 

14.91 

=  —  3.99 

341.   Leveling  for  Section. 

Leveling  for  Section  is  leveling  for  the  purpose  of 
obtaining  a  section  or  profile  of  the  surface  along  a 
given  line. 

A  Bench-mark  is  made  to  indicate  the  beginning  of 
the  line  by  drilling  a  rock  or  driving  a  nail  into  the 
upper  end  of  a  post.  Such  marks  should  be  made  at 
different  points  along  the  line,  to  serve  as  checks  in 
case  of  a  new  survey. 

It  is  necessary  also  to  measure  the  distance  between 
the  stations.  The  bearings  of  the  lines  should  be  taken 
in  case  a  map  or  plot  is  to  be  made,  representing  the 
horizontal  surface. 


350  SURVEYING. 

In  the  following  table  of  specimen  field  notes,  S.  de- 
notes stations;  B.,  bearings;  D.,  distances;  B.  S.,  back- 
sights; F.  S.j  foresights;  B.  S.  —  F.  £,  backsights  minus 
foresights;  T. D.  L.,  total  difference  of  level;  R.,  remarks, 
and  B.  M.,  bench-mark. 

The  numbers  in  the  column  headed  B.  S. — F.  S.  are 
obtained  by  subtracting  each  foresight  from  the  corre- 
sponding backsight,  observing  to  write  the  proper  sign. 

The  numbers  in  the  column  headed  T.  D.  L.  are  ob- 
tained by  continued  additions  of  the  numbers  in  the 
column  B.  S.  —  F.  S.,  each  being  the  sum  of  the  back- 
sights minus  the  sum  of  the  foresights,  up  to  a  given 
point,  expresses  the  distance  of  that  point  above  or 
below  the  bench-mark  at  the  beginning  of  the  line. 

The  minus  sign  of  a  result  indicates  that  the  sum 
of  the  foresights  exceeds  the  sum  of  the  backsights, 
and  hence,  that  the  corresponding  station  is  below  the 
first  station;  the  plus  sign  indicates  the  reverse. 

In  order  to  bring  out  prominently  the  difference  of 
level,  the  vertical  distances  are  usually  plotted  on  a 
much  larger  scale  than  the  horizontal. 

Let  us  suppose  the  numbers  in  the  column  D.  ex- 
press chains,  and  that  the  numbers  in  the  following 
columns  express  feet. 

In  the  following  profile  section  the  horizontal  dis- 
tances are  plotted  to  the  scale  of  20  chains  to  an 
inch,  and  the  vertical  distances  to  the  scale  of  20  feet 
to  an  inch. 

The  profile  of  the  section  is  therefore  distorted,  the 
vertical  distances  being  66  times  too  great  to  exhibit 
their  true  proportion  to  the  horizontal  distances. 

The  horizontal  line,  AG,  through  the  point  of  begin- 
ning is  called  the  datum  line, 


RAILROADS. 

342.   Field  Notes. 


351 


s. 

B. 

D. 

B.S. 

F.S. 

BS.—FS. 

T.D.L. 

H. 

1 

N. 

10.00 

3.25 

11.63 

—    8.38 

—  8.38 

BM.  on  post. 

2 

N. 

14.00 

4.80 

10.20 

-    5.40 

—13.78 

3 

N. 

8.25 

12.00 

1.40 

+  10.60 

—  3.18 

BM.  on  rock. 

4 

N.10°E. 

12.00 

10.80 

2.30 

-|-    8.50 

+  5.32 

5 

N.10°E. 

10.75 

1.18 

12.00 

-  10.82 

—  5.50 

6 

N. 

10.00 

2.15 

8.40 

-    6.25 

—11.75 

BM.  on  oak. 

343.    Profile  of  Section. 


JO.  oo      G 


SURVEYING  RAILROADS. 
344.    General  Plan. 

The  surveys  for  the  construction  of  railroads,  appli- 
cable also  to  canals,  graded  pikes,  dikes,  etc.,  are  made 
in  the  following  order. 

1.  The  reconnoissance,  to  locate  the  route.     The  ter- 
mini being  agreed  upon,  sometimes   several  routes  are 
examined,   so   that   an  approximate   judgment   can  be 
formed  in  reference  to  the  economy  of  construction  and 
purchasing  the  right  of  way,  the  amount  of  stock  taken 
at  different  towns  along  the  route,  and  the  profits  from 
local  business. 

2.  The    transit    survey,    to    determine    definitely  the 


352  SURVEYING. 

middle  line  along  the  surface,  after  the  route  has  been 
decided  upon  by  the  preliminary  reconnoissance. 

3.  The   section   leveling,  to  determine   the    profile   of 
the  middle  line  along  the  surface. . 

4.  The   cross-section  work,  to  determine   the  position 
and  slopes  of  the   sides,  so   that   the   amount  of  earth 
to  be  removed  or  filled  can  be  estimated. 


345.    Section  Leveling. 

Section  leveling  is  simply  an  application,  with  slight 
modifications,  of  leveling  for  section,  before  described. 

The  first  bench-mark  is  assumed  at  some  convenient 
point  near  the  beginning  of  the  line,  and  its  location 
described  in  the  column  of  remarks. 

The  datum  line  is  generally  assumed  at  such  a  depth 
below  the  first  bench-mark  —  for  example,  at  mean 
high-tide  water,  in  case  one  end  of  the  route  is  in  the 
vicinity  of  tide-water  —  that  its  whole  length  shall  be 
below  the  section  line  at  the  surface. 

The  engineer's  chain,  100  feet  in  length,  is  usually 
employed  in  taking  the  horizontal  distance. 

A  turning-point  is  a  hard  point  chosen  as  far  in 
advance  as  possible,  but  not  necessarily  in  exact  line, 
upon  which  the  rod  rests  while  a  careful  reading  is 
taken  just  before  it  is  necessary  to  change  the  position 
of  the  instrument,  whose  exact  height  above  the  datum 
line  thus  becomes  known  in  the  new  position. 

The  difference  between  a  turning-point  and  a  bench 
is  this  : 

A  turning-point  is  merely  a  temporary  point,  neither 
marked  nor  recorded,  used  to  determine  the  height  of 


RAILROADS.  353 

the   instrument   in   a   new  position.     A  bench   is  both 
marked  and  noted,  and  thus  made  permanent. 

If,  however,  it  is  thought  best  to  make  a  turning- 
point  permanent,  it  is  marked  and  recorded,  and  be- 
comes a  bench. 

In  order  that  a  bench  be  not  destroyed  in  construct- 
ing the  road,  it  should  be  a  little  removed  from  the 
line  surveyed.  The  location  of  the  benches  should  be 
carefully  noted,  so  that  they  may  be  readily  found  from 
the  field  notes. 

The  plus  sights  are  the  first  readings  of  the  rod,  made 
after  each  new  position  of  the  instrument,  as  the  rod 
stands  on  a  bench  or  turning-point,  and  are  taken  to 
thousandths  of  a  foot. 

The  minus  sights  are  the  other  readings,  and  are 
taken  to  tenths,  except  the  last  minus  sight,  before  the 
position  of  the  instrument  is  changed,  which,  being 
taken  as  the  rod  stands  on  a  turning-point  or  bench, 
is  taken  to  thousandths. 

The  height  of  the  instrument  above  the  datum  line 
is  equal  to  a  plus  sight,  plus  the  height  of  the  corre- 
sponding bench  or  turning-point. 

The  height  of  the  surfaco  above  the  datum  line,  at 
any  position  of  the  rod,  is  equal  to  the  height  of  the 
instrument,  minus  the  corresponding  backsight. 

These  heights  are  taken  at  intervals  of  1  chain,  and 
at  intermediate  points  where  the  irregularitj^  of  the 
surface  is  deemed  sufficient  to  render  it  important. 

In  the  following  field  notes  D.  denotes  distance;  B., 
bench ;  T.  P.,  turning-point ;  -f  S.9  plus  sight ;  H.  /., 
height  of  instrument ;  -  -  S,  minus  sight ;  S.  H.,  sur- 
face height;  G.  H.,  grade  height;  Cl,  cut;  F.,  fill; 
R.,  remarks. 
S.  N.  30. 


354 


SURVEYING. 

346.    Field  Notes. 


D. 

-f« 

#.  /. 

—  & 

S.H. 

G.  H. 

C. 

F. 

R. 

B. 

2.911 

32.911 

30. 

B.  50  ft. 

0. 

3.4 

29.5 

29.5 

E.  of 

1. 

4.9 

28.0 

26.5 

1.5 

0  stake. 

2. 

12.7 

20.2 

23.5 

3.3 

3. 

4.1 

28.8 

20.5 

8.3 

T.P. 

2.243 

23.755 

11.399 

21.512 

19.2 

2.3 

4. 

2.0 

21.8 

17.5 

4.3 

5. 

12.5 

11.3 

14.5 

3.2 

5.6 

4.6 

19.2 

12.7 

6.5 

0. 

12.3 

11.5 

11.5 

The  numbers  in  the  horizontal  column,  T.  P.,  are 
found  thus:  The  —  £,  11.399,  is  obtained  from  the  first 
position  of  the  instrument  by  the  reading  of  the  rod 
on  T.  P.  21.512  =  32.911  —  11.399.  The  +  £,  2.243,  is 
the  reading  of  the  rod  from  the  new  position  of  the 
instrument.  23.775  =  21.512  -f  2.243.  The  cutting  or 
filling  is  the  difference  of  S.  H.  and  G.  JI. 


347.   Profile  of  Section  and  Grade. 


32.911  above  Datum  Line. 


RAILROADS. 


355 


348.  Remarks. 

1.  The  grade  height  at  0,  minus  the  grade  at  6,  which 
is  29.5  —  11.5  ==  18  ===  the  descent  from  0  to  6.     18  -*- 
6  =  3  =  the   descent   for   1   chain,  29.5  —  3  =  26.5  = 
G.  H.  at  1 ;   26.5  -  3  =  23.5  =  G.  H.  at  2,  etc.       . 

2.  The  establishment  of  the  grade  is  influenced  by  the 
object  of  the  work,  economy,  the  balance  of  cuttings  and 
fillings,  the  points  desirable  for  termini,  etc. 

3.  The  method  exhibited  above   may  be  extended  to 
any  distance. 

349.  Example. 

Fill  out  the  notes  of  the  following  table,  and  make  a 
profile  of  section  and  grade  from  S.  H.  at  0  to  S.  H.  at  5. 


D. 

t# 

H.I. 

-S. 

S.H. 

G.Jf. 

c. 

F. 

E. 

B. 

6.248 

36.248 

30 

?L^ 

J5.  20  ft, 

0 
1 

5.3 

9.8 

IS 

n 

S.ofO. 

2 

2.3 

Sl> 

T.  P. 

10.718 

_•' 

11.814 

3 

7.6 

1%> 

4 

12.0 

13-f 

32A 

5 

2.1 

^ii 

350.    Cross-Section  Work. 

Excavations  and  embankments  are  constructed  with 
sloping  sides,  in  order  to  prevent  the  sliding  of  earth 
down  the  surface. 

The  ratio  of  slope  is  the  vertical  distance  divided 
by  the  horizontal,  and  is  therefore  the  tangent  of  the 
angle  which  the  sloping  surface  makes  with  a  hori- 
zontal plane. 

The  usual  ratio  of  slope  is  §,  and  the  angle  33°  41'. 


356  SURVEYING. 

Slope  stakes  are  driven  to  mark  where  the  sloping 
sides,  whether  of  cutting  or  filling,  will  intersect  the 
surface,  and  thus  indicate  the  boundaries  of  the  work. 

The  rod  used  in  cross-section  leveling  is  15  feet  long, 
graded  and  plainly  marked  to  feet  and  tenths,  and  is 
read  by  the  leveler  at  the  .instruments. 

The  assistants  of  the  leveler  are  the  rodman,  axman, 
and  two  tapemen. 

The  Field  book  is  ruled  into  four  columns,  headed  D. 
for  distance ;  L.  for  left ;  C.  0.  for  center-cut ;  R.  for 
right. 

The  numbers  in  the  columns  D.  and  C.  C.  are,  respect- 
ively, the  distance  and  the  corresponding  cut,  or  fill 
marked  minus  cut,  taken  from  the  field  book  for  sec- 
tion leveling. 

The  fractions  in  the  columns  L.  and  R.  have  for  their 
numerators  the  vertical  distances  of  the  cross-section, 
and  for  their  denominators,  the  corresponding  horizon- 
tal distances,  from  the  center  or  from  the  vertex  of  the 
angle  of  slope,  according  as  the  vertical  distance  is 
taken  within  or  without  the  limits  of  the  horizontal 
portion  of  the  road. 

351.   Cross-Section  Excavations. 

We  give  the  following  profile  of  cross-section,  the 
method  of  performing  the  field  operations  and  record- 
ing the  notes. 

Let  us  suppose  the  cross-section  to  be  taken  at  the 
distance  3  of  the  field  notes  of  article  343,  where  the 
center  cut  is  8.3;  that  the  road  bed  is  20  feet  wide, 
that  the  ratio  of  slope  is  §,  and  that  both  horizontal 
and  vertical  distances  are  plotted  to  the  scale  of  20 
feet  to  1  inch. 


RAILROADS. 


357 


15.7 


Take  AAf  for  the  datum  line,  and  suppose  the  read- 
ing at  the  center  stake  to  be  7.4.  The  height  of  the 
instrument  above  the  datum  line  is  therefore  8.3  + 
7.4  ±=  15.7. 

The  reading  of  the  rod  at  the  depression  F,  between 
the  center  and  the  angle  A,  is  8.5 ;  hence,  the  cut  is 
15.7—8.5  =  7.2.  The  horizontal  distance,  CF,  is  4  feet; 
hence,  the  record  in  the  field  notes,  as  seen  in  the  next 

7  2 
article  in  the  column  L,  is    j-  • 

The  reading  of  the  rod,  at  the  temporary  stake  £,  is 

o  o 

7.4;  hence,  the  cut  is  15.7  —  7.4  =  8.3,  and  the  entry,  — '^- 

A. 

The  point  S,  where  the  slope  intersects  the  surface, 
is  found  by  trial.  Since  the  vertical  distance  of  the 
slope  is  f  of  the  horizontal,  then  ES,  if  horizontal, 
would  be  |  of  EA,  which  is  12.4;  but,  on  account  of 
the  inclination  of  the  surface,  ES  will  be  less,  say  10 
feet.  Setting  the  rod  10  feet  out  from  E,  the  reading 
is  8.3,  and  hence  the  cut  =  15.7  —  8.3  ==  7.4.  Now,  f  of 
7.4  is  11.1;  hence,  the  assumed  distance,  10  feet,  is 
too  small. 

For  a  second  trial,  take  11  feet  out  from  J5",  at  which 
the  reading  of  the  rod  is  8.4,  and  the  cut  7.3.  Now,  f 
of  7.3  —  10.9,  which  lacks  but  .1  of  11,  and  is  suf- 
ficiently accurate.  The  record  for  the  slope  stake,  in 

the  column  I/,  is  -r~  • 


358  SURVEYING. 

The  reading  of  the  rod  at  the  stake  D  is  6.9;  hence, 

Q    Q 

the  cut  is  8.8,  and  the  record  in  the  column  R  is  ~  • 

A 

The  reading  at  the  elevation  G  is  5.1;  hence,  the 
cut  is  10.6.  The  horizontal  distance,  Z)G,  is  9  feet; 

,  .    10.6 
hence,  the  record  is  -^— • 

To  find  AS1'  where  the  slope  intersects  the  surface, 
since,  on  account  of  the  rising  of  the  surface,  it  is 
more  than  |-  of  8.8,  which  is  13.2,  take,  for  a  first  trial, 
18  feet  out  from  D,  at  which  point  the  reading  of  the 
rod  is  4.5,  and  hence  the  cut  15.7  —  4.5  =  11.2.  Now, 
f  of  11.2  =  16.8;  hence,  18  feet  is  too  far  out. 

For  a  second  trial,  take  17  feet  out  from  D.  The 
reading  of  the  rod  is  4.3,  and  the  cut  15.7  —  4.3  —  11.4. 
Now,  f  of  11.4  —  17.1,  which  is  sufficiently  accurate; 
hence,  the  record  for  the  slope  stake  <S",  in  the  column 

11.4 
*'  1S  171' 

'      352.   Field  Notes. 


D. 

L. 

c.  a 

R. 

3 

7.3 

8.3 
A 

7.2 
4 

\    8.3 

8.8 

10.6 
9 

11.4 
17.1 

10.9 

A' 

353.   Cross-Section  Embankments. 

The  following  is  the  profile  of  the  cross  section  drawn 
to  a  scale  of  20  feet  to  1  inch,  taken  at  the  distance 
5  of  the  field  notes  of  article  346,  where  the  filling  is 
3.2,  now  called  a  minus  cut,  and  written  —  3.2. 

Take  AA',  which  is  the  horizontal  top  of  the  embank- 
ment 20  feet  wide,  for  the  datum  line. 


RAILROADS.  359 


6.3      A  10 

" 


___ 


The  ratio  of  slope,  in  case  of  embankments,  is  — f. 

The  reading  of  the  rod  at  the  center  stake  is  6.6, 
and  the  height  of  the  instrument,  with  reference  to 
the  datum  line,  is  the  algebraic  sum  of  the  reading  of 
the  rod  and  the  minus  cut,  which  is  6.6  —  3.2  —  3.4. 

If  the  instrument  should  be  below  the  datum  line, 
the  reading  of  the  rod  would  be  numerically  less  than 
the  minus  cut,  and  the  height  of  the  instrument  would 
be  negative. 

The  readings  of  the  other  points  along  the  surface 
S&,  subtracted  from  the  height  of  the  instrument,  will 
give  the  corresponding  minus  cuts. 

The    reading    at    A    is    7.4,    the   cut,    —  4,    and   the 

record,  — -r—  - 
A 

The  reading  at  G  is  12.4,  the  cut,  — 9,  the  horizontal 

g 

distance  FG.  6.3,  and  the  record,  -mr-' 

b.o 

To  find  the  position  of  the  slope  stake  St  take  for 
the  first  trial  20  feet  out  from  F,  where  the  reading  is 
16,  and  the  cut,  —  12.6.  Now,  --  12.6  X  —  f  =  18.9; 
hence,  20  feet  is  too  far  out. 

Next  try  18  feet  out,  where  the  reading  is  15.5,  and 
the  cut,  —  12.1.  Now,  —  12.1  X  —  f  ==  18.1,  which  is 
sufficiently  accurate;  hence,  the  record  for  the  slope 

stake  S  is  — ^r  • 
lo.l 


360 


SURVEYING. 


The    reading    at    A'    is    6.4,    the    cut,   — 3,    and    the 

record,  — -p-  • 
A 

To  find  the  position  of  the  slope  stake  5",  take  for 
the  first  trial  o  feet  out  from  P,  where  the  reading  is 
6.2,  and  the  cut,  —  2.8.  Now,  --  2.8  X  --  |  ===  4.2; 
hence,  o  feet  is  too  far  out. 

Next  take  4  feet  out,  where  the  reading  is  6.1,  and 
the  cut,  —  2.7.  Now,  --2.7  X  —  f  =  4;  hence,  the 

2  7 

record  for  the  slope  stake  $'  is  - 


354.    Field  Notes. 


D. 

L. 

a  a 

R. 

5 

-12.1     -9     -4 

32 

—  3        -2.7 

18.1      6.3      A 

A'            4 

355.   Remark. 

It  sometimes  occurs  that  an  excavation  will  be  re- 
quired on  one  side,  and  an  embankment  on  the  other. 
Guided  by  the  stakes  and'  field  notes,  the  excavations 
and  embankments  can  be  correctly  made. 


356.   Computation  of  Earth-work. 

The  computation  of  earth-work  is  the  determination  of 
the  volume  of  excavation  or  embankment. 

The  cross-sections,  being  taken,  wherever  necessary, 
at  every  100  feet  or  less,  divide  the  excavations  or 
embankments  into  blocks,  which  may  be  regarded  as 
frustums  of  pyramids. 


RAILROADS. 


361 


Denoting  the  areas  of  the  sections  regarded  as  bases 
of  the  frustum  by  b  and  6',  respectively,  the  length  by 
/,  and  the  volume  by  v,  we  have  the  formula, 


357.   Examples. 

1.  The  length  of  an  excavation  is  100  feet;  find  the 
volume,  the  two  ends  being  thus  represented  : 


The  area  required,  in  each  case,  is  the  area  of  the 
whole  figure,  regarded  as  a  trapezoid,  which  is  one-half 
the  altitude  multiplied  by  the  sum  of  the  parallel  bases, 
minus  the  sum  of  the  two  triangles;  hence, 

6  =  28  X  24  —  (24  X  8  +  12  X  4)  =  432. 
V  =  19  X  12  -  (12  X  4  +    6X2)--=  168. 


v  =  J-  X  100  (432  +  168  -f  V  432  X  168). 
v  =  28980  cubic  feet  =  1073  cubic  yards. 

2.  Compute  the  volume  of  the  embankment  whose 
horizontal  breadth  at  the  top  is  16  feet,  from  the  fol- 
lowing field  notes  : 

S.  N.  31. 


362 


SURVEYING. 


D. 

L 

a  a 

R. 

5 

-11.6      -10.5 

10 

—  9.5     —8.6 

6 

17.4          A 
-17  A      -15.5 

1n 

A'          13 
-  14.2      -  13 

26.1          A 

A'          19.5 

1607  cu.  yds. 


358.    Remarks. 


1.  The    above    method   of   computing    earth-work    is 
called  by  engineers   The  mean  average  method. 

2.  The  method  known  as  The  arithmetical  mean  method 
is  easier  than  the  above,  though  less  accurate. 

The  following  is  the  formula  : 

t>  =  i*(6-h&'). 

3.  The  volume  can  also  be  computed  as  a  rectangular 
prismoid. 

4.  Irregularities   in   the   cross-section  surface  line,  as 
elevations,  depressions,  or  a  curvature  of  this  line,  must 
be  considered. 

Thus,  the  elevation 
may  be  regarded  as  a 
triangle,  its  area  com- 
puted and  added  to 

the  trapezoid   before   the   area  of  the   two  triangles   at 
the  right  and  left  be  deducted. 


359.   Railroad  Curves. 

In  the  preliminary  survey  of  a  railroad,  any  change 
in  direction  is  made  by  an  angle  which  must,  in  the 
final  survey,  be  replaced  by  a  curve,  to  which  the  sides 
of  the  angle  are  tangents. 


RAILROADS.  363 

Let   the  annexed  diagram  represent   such   an   angle 
and  curve. 

Run  out  one  of  the  tangents,  as 
BA,  to  Ej  and  let  A  denote  the  ex- 
ternal angle  EAD. 

Then  we  shall  have  C=  A,  since 
each  is  the  supplement  of  BAD, 
the  angles  B  and  D  being  right 
angles. 

Let   r  —  BC,   the   radius   of  curvature,  and  t  =  AB, 
the  tangent. 

Then,     (1)  t  =  r  tan  £  A,     (2)  r  =  rr-%-j  • 

The  degree  of  curvature  is  the  number  of  degrees  in 
an  arc  whose  length  is  1  chain  or  100  feet. 

360.    Problem. 

Given  the  degree  of  curvature,  to  find  the  radius;  and,  con- 
versely, given  the  radium  of  curvature,  to  find  the  degree. 

2  rrr  =  the  circumference, 

Q 

i)Q~  —  r^r  =  1°  of  circumference, 
— '—  —  d°  of  circumference. 


TT  1AA 

Hence,  ==  100.  18000 

'       ~" 


Having  found  the  radius  of  curvature,  we  can  find  t, 
the  tangent,  or  the  distance  from  the  vertex  of  the 
angle  to  the  point  where  the  curve  begins  by  formula 
(1)  of  the  preceding  article. 


384  SURVEYING. 

361.    Examples. 

• 

1.  Find  r  of  1°  of  curvature  and  t,  if  A  =  40°. 

Ans.  r  ==  5729.58  ft.,  t  =  2087.4  ft, 

2.  Find  r  of  2°  of  curvature  and  t,  if  A  =  40°. 

Ans.  r  =  2864.79  ft.,  t  =  1043.7  ft. 

3.  Find  r  of  3°  of  curvature  and  t,  if  A  =  50°. 

Ans.  r  =  1909.86  ft.,  t  =  890.6  ft. 

4.  Find  r  and  d,  if  A  =  35°  and  *  =  1000  ft. 

Ans.  r  =±=  3171.6  ft.,  rf  =  1°  48'  23". 

5.  Find  r  and  d,  if  A  =  100°  and  t  =  1  mile. 

.  r  =  4430.4  ft.,  d  =  1°  17'  35". 


362.   Location  of  the  Curve. 

Method. 


Let  each  of  the  arcs,  £p,  pq,  qr,  ...  be  1  chain,  then 
will  the  number  of  degrees  in 
each,  or  in  the  corresponding 
angle  at  the  center,  be  equal 
to  d,  the  degree  of  curvature. 

The  angle  ABp,  formed  by 
a  tangent  and  a  chord,  is 
measured  by  one-half  the  arc 
Bp,  and  is  therefore  equal  to 

i^ 

Each  of  the  inscribed  angles, 
pBq,  qBr,  is  measured  by  one-half  the  intercepted  arc, 
and  is  therefore  equal  to  ^d. 

Having  determined  the  point  B,  where  the  curve 
begins,  the  transitman  sets  his  instrument  at  this 
point,  and  directs  it  to  A.  He  then  turns  it  an  angle 
equal  to  JrZ,  on  the  side  toward  the  curve. 


RAILROADS.  365 

The  chainmen  then  take  the  chain,  the  follower 
placing  his  end  at  B,  and  the  leader  drawing  out"the 
chain  at  full  length  toward  ^4,  is  directed  by  the  trans- 
itman  into  line  so  as  to  locate  the  -point  p,  at  which 
the  axman  drives  a  stake. 

The  transitman  again  turns  his  instrument  an  angle 
equal  to  Jrf,  the  chainmen  advance,  the  follower  plac- 
ing his  end  of  the  chain  at  p,  the  leader  again  draw- 
ing out  the  chain  at  full  length,  is  directed  by  the 
transitman  so  as  to  locate  the  point  (/,  at  which  the 
axman  drives  a  stake,  and  so  on. 

The  last  distance  will  usually  not  be  1  chain;  but  if 
?i  be  the  number  of  preceding  deflections,  the  last  angle 
of  deflection,  since  the  sum  of  all  the  deflections  is  equal 
to  \C  =  T.  A,  will  be  equal  to 

I  A  —  \dn. 

It  is  to  be  observed  that  the  chord  is  made  equal  to  1 
chain  instead  of  the  arc;  but  as  the  radius  is  much 
greater  than  the  chord,  the  arc  and  chord  will  not  differ 
materially,  and  no  appreciable  error  arises  in  practice. 

Second  Method. 

Points  on  the  curve  may  be  located  by  the  use  of  two 
transits,  without  the  use  of  the  A 

chain,  as  may  be  desirable,  in 
case  the  curve  is  to  be  located 
in  marsh}7  ground  or  shallow 
water. 

•?> 

Let  one  transit  be  placed  at 
B  and  another  at  /),  the  extremities  of  the  curve. 

Direct  the  transit  at  B  to  A,  the  one  at  D  to  B,  then 
turn  each  to  the  right  an  angle  equal  to  ^d°. 


366 


SURVEYING. 


The  intersection  of  the  lines  will  -determine  _p,  where 
the  axman,  directed  by  both  transitmen,  drives  a  stake. 

.In  like  manner  other  points  can  be  located. 

If  A  is  visible  from  D,  but  not  B,  direct  the  transit 
at  D  to  A;  then,  to  locate  p,  turn  it  to  the  left  an  angle 
equal  to  \A°  —  Jd°. 

To  locate  7,  turn  the  transit  at  D  from  p  to  the 
right  an  angle  equal  to  £d°,  or  from  A  to  the  left  an 
angle  equal  to  \A°  —  d°,  and  the  transit  at  B  to  the 
right  from  p  an  angle  equal  to  -Jd°,  or  to  the  right 
from  A  an  angle  equal  to  cZ°,  and  so  on. 


Third  Method. 

Let  B  be  the  point  where  the  curve  begins.  Take 
Bm  equal  to  1  chain.  Then, 
to  find  the  length  of  the  off- 
set mp,  complete  the  circle, 
draw  the  diameter  BE,  let 
fall  the  perpendicular  pn  to 
BE,  and  draw  pE. 

In  the  right  triangle  BpE, 
Bp  is  a  mean  proportional 
between  BE  and  Bn  ;  hence, 
BExBn^Ttp1*;  but  BE  = 
2  r,  Bp  =  l,  and  Bn  =  mp, 


To  find  q,  produce  Bp  till  ps  =  1  chain,  and  draw  tv, 
tangent  to  the  curve  at  p. 

Then,  *pv  =  tpB  =  mBp  =--  ypq, 

For  the  first  and  second   are  vertical,  and   all   the  rest 
are  included  between  tangents  and  equal  chords. 


RAILROADS. 


367 


.  • .  spq  =  2  mBp,    . ' .  the  arc  sq  =  2  arc  mp, 

Or,  the  arcs  being  small,  do  not  differ  materially  from 
their  chords, 

.  • .    sq  =  2  mp  = 

Hence,  to  locate  a  curve  by  this  method  without  the 
transit,  commence  at  B,  where  the  curve  is  to  begin, 
take  Bm  =  1  chain  in  the  direction  of  the  straight 

track,  make  the  offset  mp  =  — ,  produce  Bp  till  ps  =  1 

chain,  make  the  offset  sq  equal  to  twice  the  first  offset, 
produce  pq  till  the  produced  part  —  1  chain,  make  an 
offset  equal  to  the  last,  and  so  on. 

Fourth  Method. 
It  is  evident  from  the  diagram  that 


But  BC  =  r,  and  nC  =  Vr*  —  t2. 


.  • .     mp  =  r  —  1   r2  —  t2. 

By  giving  to  t  different  values,  other 
points  of  the  curve  can  be  determined. 

Fifth  Method,. 
It  is  evident  from  the  diagram  that 

mp  =  mC —  Cp. 
But  mC  =  Vr2  -\-  t2,  and  Cp  —  r. 


•'•     ™>P  =  Vr2  -\-t2—r. 

In    this    method    the    offset    is   not 
made  at  right  angles  to  the  tangent, 
but  in  a  direction  toward  the  center,  which  is  supposed 
to  be  visible  from  m. 


368 


SURVEYING. 


The  preceding  methods  apply  to  points  of  the  curve 
1  chain  or  100  feet  from  each  other,  which  will  be 
sufficient  for  the  excavations  or  embankments. 

Before  laying  the  track,  stakes  are  driven  at  points 
on  the  curve,  distant  from  each  other  about  10  feet. 


363.    Problem. 

To  locate  inter  mediate  points  on  the  curve. 

Let   the   diameter  in  the  diagram  be  parallel  to  the 
chord,  which  is  equal  to  1 
chain  =  100  feet,  the  ordi- 
nates  a,  6,  e,  el,  e,  /,  e.  d,  c,  fr,  a 
be  10  feet  from  each  other,        / 
and  v,  w,  or,  y,  z,  y,  x,  w,  v  be       / 
offsets  from  the  chord  to  the 
curve,  corresponding  to  the  ordinates  b,  c,  d,  e,f,  e,  d,  c,  b. 

The  square  of  an  ordinate  is  equal   to  the  rectangle 
of  the  segments  into  which  it  divides  the  diameter. 


a*  =         (r  —  50)  (r  +  50),     a  =  1  \r  —  50)  (r 
b  =  [   (r  —  40)(r  +  40), 


c  =      (r  — 


—-  c —  a. 


d  =  l   (r  — 20)  (r-H  20),     x  =d—a. 

e  =  \   (r— T.O)  (r~4^10),     y  =  e  —  a. 
f=r,  z  =  f  —  a. 


364.    Example. 


Find   the   radius  of  a   1°  curvature,  and  the   offsets 
from  the  chord  of  100  feet  to  the  curve. 

=  5729.58  ft.,  v  ==  .08  ft.,  w  -=  .14  ft. 
=  .19  ft.,  y  =  .21  ft.,  z  —.22  ft. 


(r  = 
Ans.  | 


TOPOGRAPHICAL. 

TOPOGRAPHICAL   SURVEYING. 

365.    Definition  and  Method. 

Topographical  surveying  is  that  branch  in  which  the 
form  of  the  surface,  the  situation  of  ponds,  streams, 
marshes,  rocks,  trees,  buildings,  etc.,  are  considered 
and  delineated. 

The  surface  is  supposed  to  be  intersected  by  hori- 
zontal planes  equally  distant  from  each  other,  and  the 
curves  formed  b}^  the.  "intersection  of  the  planes  and 
the  surface  projected  on  a  horizontal  plane. 

These  projections  will  be  nearer  together  or  farther 
apart,  according  as  the  slope  of  the  surface  approaches 
a  vertical  or  a  horizontal  position. 

The  operations  are  of  two  kinds — field  operations  and 
plotting. 

366.   Field  Operations. 

Field  operations  consist  in  finding  and  recording 
points  of  the  curves  of  intersection  of  the  surface  and 
the  horizontal  planes,  the  course  of  streams,  and  the 
situation  of  noteworthy  objects  on  the  surface. 

Range  with  the  level,  or  transit  theodolite,  which 
is  more  convenient  in  topographical  operations,  stakes 
marked  as  in  the  annexed  diagram,  and  cause  them 
to  be  driven  into  the  ground,  at  a  horizontal  distance 
from  each  other  of  100  feet  or  less,  varying  with  the 
inequality  of  the  surface  and  the  degree  of  accuracy 
with  which  it  is  desirable  that  the  work  be  executed. 

Find  by  the  eye,  or  by  the  instrument  if  necessary, 
the  lowest  point  in  the  field,  at  which  make  a  permanent 
bench-mark,  and  assume  for  the  plane  of  reference  the 


370 


SURVEYING. 


BI 

S 

1)3          D* 

c, 

oa 

C3            C, 

B3            B4 

S' 

A, 

*! 

A.,             A4 

horizontal  plane  passing  through  this  point,  which  we 
will  suppose  to  be  C\. 

Place  the  instrument  at 
some  convenient  station,  S, 
from  which  take  the  read- 
ing of  the  rod  at  Cr1?  which 
suppose  to  he  10.378,  and 
enter  this  as  a  backsight  in 
the  field  notes. 

Take  the  readings  of  the 
rod  at  as  many  stakes  as 
possible  from  the  station  S.  Suppose  these  readings  to 
be  <72,  6.481;  C8,  1.214;  Z>1?  8.235;  D2,  6.378;  D3,  4.102; 
D4,  2.304,  and  enter  these  readings  in  the  field  notes 
as  foresights,  placing  the  smallest  reading,  (73,  last. 

At  (73  drive  a  small  stake  for  a  check. 

Subtract  the  foresight  O2  6.481  from  the  backsight 
10.378,  and  enter  the  difference  in  the  column  of  differ- 
ence, headed  D.;  also  in  the  column  of  total  difference 
of  level  above  C,,  headed  T.  D.  L. 

Subtract  each  of  the  remaining  foresights  from  the 
next  preceding  one,  and  enter  the  results,  with  their 
proper  signs,  in  the  column  D. 

Add  each  result  to  the  previous  total  difference  of 
level,  and  enter  the  results  in  the  column  T.  D.  L. 

The  total  difference  of  level  for  C3  is  also  found  by 
subtracting  the  foresight  of  O3  from  the  backsight  of 
C15  which,  compared  with  the  result  before  found,  will 
serve  as  a  cheek. 

Move  the  instrument  to  S",  and  take  a  backsight  to 
the  check  stake  O3,  and  the  foresights  to  as  many  of 
the  remaining  stakes  as  possible,  suppose  all  of  them 
and  enter  the  readings  in  the  field  notes  as  before. 


TOPO  GRA  PHICA  L. 


871 


Subtract  the  first  of  these  foresights  from  the  back- 
sight (?3,  and  add  the  result  to  the  total  difference  of 
level  for  (73,  and  enter  the  sum  in  the  column  T.  D.  L. 

Subtract  each  of  the  following  foresights  from  the 
next  preceding  foresight,  and  enter  the  result,  with  its 
proper  sign,  in  the  column  ZA,  and  add  it  to  the  next 
preceding  difference  of  level,  and  enter  the  sum  in 
the  column  T.  I).  L. 

As  a  check,  subtract  the  foresight  of  Z?3  from  the 
backsight  O3;  the  difference  will  be  the  height  of  Z?3 
above  C3,  which  add  to  the  former  check  number, 
which  is  the  difference  of  level  of  <73  and  C^,  and  the 
sum  will  be  the  total  difference  of  level  of  Bs  and  Cl. 

Compare  the  explanations  of  this  article  with  the 
field  notes  of  the  following  article. 

367.    Field  Notes. 


B.  S, 

F.  S. 

D. 

T.  I).  L. 

R. 

P\ 

0.000 

Ci 

10.378 

C2 

6.481 

+  3.897 

C2 

3.897 

DI 

8.235 

-  1.754 

D, 

2.143 

D2 

6.378 

+  1.857 

D2 

4.000 

D3 

4.102 

+  2.276 

D3 

6.276 

#4 

2.304 

+  1.798 

D* 

8.074 

c, 

1.214 

+  1.090    |   C3 

9.164 

Check  9.164 

C3 

9.687 

a< 

12.000 

-  2.313 

c* 

6.851 

B, 

11.845 

+  0.155 

B, 

7.006 

B2 

5.184 

+  6.661 

B2 

13.667 

B4 

8.314 

-  3.130      B.v 

10.537 

AI 

12.000 

—  3.686 

Al 

6.851 

Az 

11.321 

+  0.679 

A, 

7.530 

A3 

10.987 

+  0.334 

A3 

7.864 

A4 

7.125 

+  3.862 

A, 

11.726 

B3 

0.132 

4-  6.993 

B3 

18.719 

Check  9.555 

18.719 

372 


SURVEYING. 


308.    Plotting. 

Let  the  annexed  diagram  be  a  plot  of  the  ground  on 
which  is  written,  with  red  ink,  the  height  to  tenths, 
taken  from  the  field  notes,  of  the  surface,  at  each  stake, 
above  the  plane  of  reference  passing  througli  C1. 

Let  us  suppose  that  the 
horizontal  planes  intersect- 
ing the  surface  are  4  feet 
apart. 

The  intersection  of  the 
surface  and  the  plane  4  feet 
above  the  plane  of  refer- 
ence crosses  the  line  Al  Dl 
between  the  points  Bl  C19 
at  a  point  4  feet  above  C^. 

To  determine  this  point,  observe  that  the  rise  from 
Cl  to  Bl  is  7  feet.  Then  the  distance  on  this -line  from 
C\  to  the  point  where  the  height  above  (7t  is  4  feet  is 
found  by  the  proportion, 


:  4  : :   100  :  x, 


x  —-  57.1. 


This  method  assumes  the  ascent  to  be  uniform  be- 
tween Bl  and  C^;  but  this  point  can  be  tested  and 
other  points  of  the  curve  found  as  follows :  Set  up  the 
instrument  at  S,  and  make  the  backsight  to  C\  10.378, 
the  same  as  before ;  then  depress  the  vane  on  the  rod 
4  feet  —  that  is,  to  the  reading  6.378. 

Now  let  the  rodman  set  up  the  rod  at  the  point  be- 
tween Cx  and  Bl  determined  from  the  proportion,  and 
let  the  surveyor  observe  whether  the  horizontal  wire 
of  the  telescope  ranges  with  the  horizontal  line  of  the 
vane ;  if  not,  let  the  rod  be  moved  a  little  toward  B^  or 


TO  PO  G  RA  PHICA  L. 


373 


C1  till  they  do  range,  and  at  the  point  thus  determined 
let  a  stake  marked  4  be  driven  by  the  axman. 

An  inspection  of  the  plot  will  show  that  the  curve 
passes  between  B2  and  C2  at  a  distance  from  C2  found 
from  the  proportion, 

9.8  :  .1   ::   100  :  a-,     .-.  sf=l. 

Let  the  rodman  advance  toward  this  point,  pausing 
at  one  or  two  intermediate  points,  and  at  this  point, 
whose  positions  are  definitely  determined  and  marked. 

In  a  similar  manner  determine  where  the  curve 
crosses  (72  £3  and  trace  it  to  Z)2. 

In  like  manner,  trace  the  curves  of  intersection  of 
the  surface  and  planes,  8  feet,  12  feet,  and  16  feet  above 
the  plane  of  reference,  and  let  these  curves  be  marked 
on  the  ground  by  stakes  numbered  8,  12,  and  16,  re- 
spectively. 

The  horizontal  distance  of  each  stake  from  two  sides 
of  a  square  can  be  measured  and  recorded.  From  this 
record  the  surveyor  can  draw  the  curves  on  the  plot 
as  exhibited  above. 


3G9.   Shading. 


The  slopes  may  be  repre- 
sented to  the  eye  by  short 
lines  drawn  perpendicular 
to  the  curves,  marking  the 
intersection  of  the  surface 
with  the  horizontal  planes. 
These  lines  are  heaviest  and 
closest  where  the  slopes  are 
steepest,  and  lighter  where 
the  slopes  are  less  abrupt. 


374 


SURVEYING. 


370.   Conventional  Signs. 

The  following  conventional,  though  not  altogether 
arbitrary  signs,  are  used  to  indicate  objects  worthy 
of  note  : 


Pasture. 


Sand. 


Gardens. 


Meadow. 


Fields. 


Orchards.  Swamp. 

•'••:••:•:•    ftifiift    -a^i^i^^: 


Cotton. 


Turnpike, 
Common  Road, 
Toot  Path, 
Rail  Road,     mum 

Stone  Bridge, 
Suspension  Bridge, 

Carriage  Ford, 
Canal  &  Lock, 

D$  Water  Mill. 

D /"Steam  Mill. 

^>  Post  Office. 

ti  Hotel. 


Bushes. 


•I 


•  1  Railroad  Station. 

$  Telegraph  Station. 

6  Church. 

A  Monument. 
X  Custom  House,  i  Way  mark. 
O  Building,  Wood.  £  Mile  Stone. 
m         "       Stone.  J^  Lime  Kiln. 
jv-lGold.  y   Silver, 

if    Tin.  -^  Lead. 


Vineyard.  Hopfield. 


Hedges, 

Rail  &nC 

Boardfence, 
Stone  Wall, 


Wootl  Bridge, 
Pontoon  Bridge, 

Horse  Ford, 

Stone  Dam. 

•$     Laud-mark,  stone.  0  Light-house,  rev. 

?         "        "      wood.  >&c     -"         "     fixed. 

^^       "        "    mouud.  J.tX  Beacons. 

^  trees,  ^-to   Ancliorago. 

^  Survey  Station.  P    ra  Buoys. 

€>  Rock  bare.  < — 1«««  Current. 

*i|?'Sunken  rocks.  ^^  Nb  Current. 

9    Copper.  C?  Iron, 

$    Mercury.  •  Coal. 


BAROMETRIC  HEIGHTS.  375 

371.    Finishing  a  Map. 

The  points  of  compass  are  indicated  as  is  usual,  the 
top  of  the  map  denoting  the  north,  etc.,  etc. 

The  meridian,  both  true  and  magnetic,  should  be 
drawn,  and  the  variation  of  the  needle  indicated. 

The  lettering  should  be  executed  with  care,  after 
printed  models  of  various  styles. 

The  border  may  be  made  by  a  heavy  line,  relieved 
by  a  light  parallel. 

The  title,  in  ornamental  letters,  should  occupy  one 
corner  of  the  map,  with  the  name  of  the  locality,  the 
dates  of  the  survey  and  drawing,  and  the  names  of 
the  surveyor  and  draughtsman. 

The  scale  of  horizontal  distances,  for  finding  and  com- 
paring distances  on  the  map,  and  the  scale  of  construction, 
used  in  the  smallest  measurements  required  in  project- 
ing dimensions  in  the  drawing,  should  be  accurately 
drawn  in  some  convenient  position  within  the  border. 

Parallels  of  latitudes  and  meridians,  in  extended  sur- 
veys, should  be  drawn  in  their  true  position. 

BAROMETRIC  HEIGHTS. 

372.    Preliminary  Remarks. 

The  barometer  affords  an  approximative  method  for 
finding  the  difference  of  level  of  two  stations. 

To  attain  to  as  great  a  degree  of  accuracy  as  possible, 
it  is  important  to  employ  two  good  barometers,  one  at 
the  lower  and  the  other  at  the  upper  station. 

Before  using  the  barometers,  they  should  be  carefully 
compared  by  frequent  trials,  and  the  variation  ascer- 
tained, which  is  to  be  allowed  for  in  the  observations. 


376  SURVEYING. 

Increased  accuracy  is  attained  by  making  repeated 
observations,  and  taking  the  mean  of  the  results. 

To  guard  against  varying  local  conditions  of  the  at- 
mosphere affecting  pressure,  beside  difference  of  eleva- 
tion, the  stations  should  not  be  distant  from  each  other 
more  than  four  or  five  miles;  and  the  observation^ 
should  be  made  when  there  is  no  wind. 


373.   Bailey's  Formula. 

The  subjoined  formula  requires  a  knowledge,  at  both 
stations,  of  the  height  of  the  column  of  mercury,  its 
temperature  as  indicated  by  an  attached  thermometer, 
the  temperature  of  the  air  as  indicated  by  a  detached 
thermometer,  and  the  latitude  of  the  locality. 

Let  d  denote  the  difference  of  level  in  feet; 
/,  the  latitude  of  the  place  in  degrees; 

A,  T,  t,  respectively,  the  height  of  the  barometer,  the 
temperature  of  the  mercury,  and  the  temperature  of 
the  air  at  the  lower  station ; 

A',  T",  t',  respectively,  the  same  at  the  upper  station. 

Then,  d  =  60345.51  [1  +  .001111  (t  +  *'  —  64)] 

h 


X  (1  +  .002695  cos  2  /)  X  log  77 


Let  ,4=,  log  560345.51  [1+.  001111  (*  +  *'  —  64;]}, 

.  B  =  log  (1  -f-  .002695  cos  2  I  ), 

c  =  log  [i  -f  .0001  (T—  r  )], 

D=log  A-—  (log  A'-f<7). 
.-.     log  d  =  A+  £  +  log  D. 


This  formula  is  applied  by  the  aid  of  the  tables: 


BAROMETRIC  HEIGHTS. 


377 


374.    Howlet's  Tables. 

Table  A,  for  Detached   Thermometer. 


t  +  r 

A. 

t  i  £ 

A. 

t  +  tf 

A. 

t  +  r 

A. 

1° 

4.74914 

46° 

4.77187 

91° 

4.79348 

136° 

4.81407 

2° 

.74966 

47° 

.77236 

92° 

.79395 

137° 

.81452 

3° 

.75017 

48° 

.77285 

93° 

.79442 

138° 

.81496 

4° 

.75069 

49° 

.77335 

94° 

.79489 

139° 

.81541 

5° 

.75120 

50° 

.77384 

95° 

.79535 

140° 

.81585 

6° 

.75172 

51° 

.77433 

96° 

.79582 

141° 

.81630 

7° 

.75223 

52° 

.77482 

97° 

.79628 

142° 

.81674 

8° 

.75274 

53° 

.77530 

98° 

.79675 

143° 

.81719 

9° 

.75326 

54° 

.77579 

99° 

.79721 

144° 

.81763 

10° 

.75377 

55° 

.77628 

100° 

.79768 

145° 

.81807 

:  11° 

.75428 

56° 

.77677 

101° 

.79814 

146° 

.81851 

12° 

.75479 

57° 

.77725 

102° 

.79861 

147° 

.81896 

13° 

.75531 

58° 

.77774 

103° 

.79907 

148° 

.81940 

14° 

.75582 

59° 

.77823 

104° 

.79953 

149° 

.81984 

15° 

.75633 

60° 

.77871 

105° 

.79999 

150° 

.82028 

16° 

.75684 

61° 

.77919 

106° 

.80045 

151° 

.82072 

17° 

.75735 

62° 

.77968 

107° 

.80091 

152° 

.82116 

18° 

.75786 

63° 

.78016 

108° 

.80137 

153° 

.82160 

19° 

.75837 

64° 

.78065 

109° 

.80183 

154° 

.82204 

20° 

.75888 

65° 

.78113 

110° 

.80229 

155° 

.82248 

21° 

.75938 

66° 

.78161 

111° 

.80275 

156° 

.82291 

22° 

.75989 

67° 

.78209 

112° 

.80321 

157° 

.82335 

23° 

.76039 

68° 

.78257 

113° 

.80367 

158° 

.82379 

24° 

.76090 

69° 

.78305 

114° 

.80413 

159° 

.82423 

25° 

.76140 

70° 

.78353 

115° 

.80458 

160° 

.82466 

26° 

.76190 

71° 

.78401 

116° 

.80504 

161° 

.82510 

27° 

.76241 

72° 

.78449 

117° 

.80550 

162° 

.82553 

28° 

.76291 

73° 

.78497 

118° 

.80595 

163° 

.82597  I 

29° 

.76342 

74° 

.78544 

119° 

.80641 

164° 

.82640 

30° 

.76392 

75° 

.78592 

120° 

.80686 

.  165° 

.82684 

31° 

.76442 

76° 

.78640 

121° 

.80731 

166° 

.82727 

32° 

.76492 

77° 

.78687 

122° 

.80777 

167° 

.82770 

33° 

.76542 

78° 

.78735 

123° 

.80822 

168° 

.82814 

34° 

.76592 

79° 

.78782 

124° 

.80867 

169° 

.82857 

35° 

.76642 

80° 

.78830 

125° 

.80913 

170° 

.82900  ! 

36° 

.76692 

81° 

.78877 

126° 

.80958 

171° 

.82943 

37° 

.76742 

82° 

.78925 

127° 

.81003 

172° 

.82986 

38° 

.76792 

83° 

.78972 

128° 

.81048  1 

173° 

.83029 

39° 

.76842 

84° 

.79019 

129° 

.81093  ! 

174° 

.83072 

40° 

.76891 

85° 

.79066 

130° 

.81138 

175° 

.83115 

41° 

.76940 

86° 

.79113 

131° 

.81183 

176° 

.83158 

42° 

.76990 

87° 

.79160 

132° 

.81228 

177° 

.83201 

43° 

.77039 

88° 

.79207 

133° 

.81273 

178° 

.83244 

44° 

.77089 

89° 

.79254 

134° 

.81317 

179° 

.83287 

45° 

.77138 

90° 

.79301  ! 

135° 

.81362 

180° 

.83329 

378 


SURVEYING* 


Table  B,  for  Latitude. 


I. 

B. 

1.     B.     /.     B. 

/.     B. 

0° 

0.00117 

27°  0.00069   50° 

1.99980 

59° 

1.99945 

3° 

.00116 

30° 

.00058   51°  ,  .99976 

60° 

.99941 

6° 

.00114 

333 

.00048   52°  i  .99972 

63°   .99931 

9° 

.00111 

36°  '  .00036   53°  i  .99968 

66° 

.99922 

12° 

.00107 

39  D  !  .00024   54° 

.99964 

69° 

.99913 

15° 

.00101 

42° 

.00012  i  55° 

.99960 

75° 

.99899 

18° 

.00095 

453 

.00000  i  56° 

.99956 

80°  j  .99890 

21° 

.00087 

48  D 

1.99988  i  57° 

.99952 

85° 

.99885  ! 

24° 

.00078 

49° 

.99984  :  58° 

.99949 

90° 

.99883 

Table  C,  for  an  Attached  Thermometer. 


T—T/ 

C. 

T—T' 

C. 

T—  T'\   C. 

T—  T' 

a 

0° 

0.00000 

12' 

0.00052 

24°  0.00104 

36° 

0.00156 

1° 

.00004 

13° 

.00056 

25°  !  .00108 

37° 

.00161 

2° 

.00009 

14° 

.00061 

26°  ,  .00113 

38° 

.00165 

3° 

.00013 

15° 

.00065 

27°  I  .00117 

39° 

.00169 

4° 

.00017 

16° 

.00069 

28°  !  .00121 

40° 

.00174 

5° 

.00022 

173 

.00074 

29°   .00126 

41° 

.00178 

6° 

.00026 

18° 

.00078 

30°  !  .00130 

42° 

.00182 

7° 

.00030 

19° 

.00082 

31°  ,  .00134 

43° 

.00187 

8° 

.00035 

20° 

.00087 

32°  |  .00139 

44° 

.00191 

9° 

.00039 

21° 

.00091 

33°  !  .00143 

45° 

.00195 

10° 

.00043 

22° 

.00095 

34°  1  .00148 

46° 

.00200 

11° 

.00048 

23° 

.00100 

35°   .00152 

47° 

.00204 

375.    Examples. 

1.  At  the  mountain  Guanaxuato,  in  Mexico,  lat.  21°, 
Humboldt  made  the  following  observations: 

Lower  Station.  Upper  Station. 

Barometric  column,           h  ==  30.05,  h'   =  23.66. 

Attached  thermometer,     T  =  77°.6,  T'  =  70°.4. 

Detached  thermometer,    t   =  77°. 6,  t'    =  70°.4. 

log  d  =  A  +  B  4-  log  D. 


BAROMETRIC  HEIGHTS.  379 

log  h  (30.05)  =5=  1.47784  A  =  4.81940 

log  A' (23.66)  ==  1737402  B  =  0.00087 

Table  C  gives  C  =  0.0003.1  log.  D  =  101498 

log  A'+C  ==  L37433  log.  d  =  3^83525 

7)  —  log  A  —  (log  h'-}-  C)  ===  0.10351  .  • .    d  ==  6843  ft. 

„  2.  Find  the  difference  of  level  of  two  stations,  lat.  42°, 
from  the  following  data: 

h  =  30,    T  ==  75°.o,    t  =  75°.  1 

#  =  25,    7"  =  70°.3,   *'  =  70°.  j    ^71S'  5195'  ftl 

3.  Find  the  difference  of  level  of  two  stations,  lat.  45°, 
from  the  following  data : 

h  =  29.2,    T  =  80°.3,   t  =  80°. 


i'  =  27.1,    T  fcft;  77°.4,   t'  =  77°.  J    ~'"*'  2149'9 
4.  Find  d,  lat.  50°,  from  the  following  data  : 
h  —29,    T  ==  60°.  1,   f  —  60°. 


'=  28,    r=  59°M,   t'=      °  '  973'8 


370.    Leveling  with  one  Barometer. 

Take  the  observations  at  the  lower  station,  then  pro- 
ceed to  the  upper  station  and  take  the  observations 
there,  and  note  the  interval  of  time  which  has  inter- 
vened, then  go  back  to  the  lower  station  and  at  the 
expiration  of  an  equal  interval  repeat  the  observations. 

Reduce  the  mercurial  column  of  the  second  observa- 
tion at  the  lower  station  to  what  it  would  have  been 
at  the  temperature  of  the  first  observation,  on  the 
principle  that  mercury  expands  or  contracts  .0001  of  its 
volume  for  each  degree  of  increase  or  diminution  of 
temperature. 

Then  take  the  arithmetical  mean  of  this  reduced 
height  and  the  first  observed  height  for  the  height  at 
the  lower  station,  the  mean  of  the  temperature  denoted 


380  SURVEYING. 

by  the  detached  thermometer  at  the  lower  station  for 
the  temperature  of  the  air  at  thai;  station,  and  the 
temperature  denoted  by  the  attached  thermometer  at 
the  first  observation  for  the  temperature  of  the  mer- 
cury, then  proceed  as  if  the  observations  had  been  taken 
with  two  barometers. 

377.    Examples. 

(  1st  obv.,  h  =  29.62,  T  =  56°,5,  t  ==  56°. 

1.  J  Lower  sta-  \  2d  obv.,  h  r—  29.63,  T  =  63°,     t  =  61°. 
\Lat.  41°.4;  upper  sta.,  A'—  28.94,  7"=570.5,  t'=57°. 

Reducing  h  of  2d  obv.  from  T=  63°  to  7^56°.5,  we  have, 
Reduced  h  =  29.63  (1  —  6,5  X  .0001)  ==,  29.611. 

,        29.62  +  29.611 
.  •  .     Mean        h  -  —  -  —  =  29.6155. 

Mean        t  l=  56°  +  61°    =  58°.5. 

.-.    t  +  t'  =  58°.5+57°       =  :  115°.5. 
andT  —  r=--56°.5  —  57°.5  =    -1°. 

log  h  (29.6155)  =  1.47152           A  =  4.80481 

log  h'  (28.94)  L46150           B  —  0.00014 

C  =  -0.00004  log  D  =  2J00260 

logA'+C=  i"46146  log  d  =  2.80755 

D  =  \ogh—  (log  A'+C)=  0.01006'  .-.  d  ==   642  feet. 

f  1st  obv.,  A  =  29.7,  71  ==  60°,    t  =  60°. 

2.  J  Lower  sta>    I  2d  obv.,  h  =  29.75,   T  =  66°,   t  =  66°. 
\Lat.  40°;    upper  sta.  &'=  28.6,  '  T'=  62°,  -  f=  62°. 

i=  1077  ft. 


(  /  1st-  obv.,   h  ---  29.6,     7r  =  50°,    «  =  50°. 

3.  )Lowersta-   \2d  obv.,   A  =  29.65,  T  =  46°,    Z  =  46°. 

(Lat.  50°;    upper  sta.    A'=  27.6,     T'^45°,    *'=  45°. 

Ans.  d  =  1909  ft. 


NAVIGATION. 

PRELIMINARIES. 

378.    Definition  and  Classification. 

Navigation  is  the  art  of  ascertaining  the  place  of  a 
ship  at  sea,  and  of  conducting  it  from  port  to  port. 

There  are  two  methods  of  finding  the  place  of  a  ship: 

1.  By  dead   reckoning;   that   is,  by  tracing   from   the 
record  the  courses  and  distances  sailed. 

2.  By  Nautical  Astronomy;  that  is,  by  deducing  the 
latitude  and   longitude  of  the  place  of   the  ship  from 
celestial  observations. 

The  first  method  is  subdivided  into  the  following: 

Plane  sailing,  parallel  sailing,  middle  latitude  sailing, 
Mercator's  sailing,  and  current  sailing. 

379.   The  Mariner's  Compass. 

The  magnetic  needle  rests  on  a  pivot,  so  as  to  turn 
freely. 

The  compass  box  is  suspended  by  gimbals  or  rings, 
turning  on  axes  at  right  angles  to  each  other,  thus 
securing  a  horizontal  position  notwithstanding  the  roll- 
ing motion  of  the  ship. 

A  circular  card,  whose  circumference  is  divided  into 
thirty-two  equal  parts,  called  points,  each  of  which  is 

(381) 


382 


NAVIGATION. 


subdivided  into  four  equal  parts,  called  quarter  points, 
rests  upon  the  needle,  with  which  it  turns  freely. 


N.  b.  E.  is  read  north  by  east ;   N.  N.  £".,  north  north- 
east, etc. 

380.    Table  of  Points  and  Angles. 


• 

North. 

South. 

Angles. 

1 

N.b.E. 

N.b.W. 

S.b.E. 

S.b.W. 

11°  15' 

2 

N.N..E. 

N.N.W. 

S.S.E. 

s.s.w. 

22°  30' 

3 

N  E.b.N. 

N.W.b.N. 

S.E.b.S. 

S.W.b.S.  i   33°  45' 

4 

N.E. 

N.W. 

S.E. 

S.W.        45°  (X 

5 

N.E.b.E.  N.W.b.W. 

S.E.b.E. 

S.W.b.W.    56°  15' 

6 

E.N.E. 

W.N.W. 

E.S.E. 

W.S.W.      67°  30' 

7 

E.b.N. 

W.b.N. 

E.b.S. 

W.b.S.       78°  45' 

8 

E. 

W. 

E. 

W.          90°  0' 

Note  1.— J  point  =  2°  48'f,  J  point  =  5°  37'^,  i  point 
=  8°  26'}. 

Note  2. —  The  compass  is  placed  near  the  helm,  at  the 
stern,  and  the  line  from  the  center  of  the  compass  to 
the  ship's  head  indicates  the  track  of  the  ship. 


PRELIMINARIES.  383 

381.   Variation  and  Deviation  of  the  Compass. 

The  variation  of  the  compass  is  the  angle  included  be- 
tween the  magnetic  meridian  and  the  true  meridian. 

The  amount  of  variation  is  ascertained  by  Nautical 
Astronomy. 

The  deviation  of  the  compass  is  the  deflection  of  the 
needle  from  the  magnetic  meridian,  caused  by  the  iron 
in  the  ship. 

The  amount  of  deviation  is  ascertained  by  special 
experiments. 

382.   Course,  Leeway,  Rhumb  Line. 

The  compass  course  of  a  ship,  at  any  point,  is  the 
angle  which  her  track  makes  with  the  magnetic  me- 
ridian at  that  point. 

The  true  course  of  a  ship,  at  any  point,  is  the  angle 
which  her  track  makes  with  the  true  meridian  at  that 
point. 

In  the  compass  course,  the  deviation  is  supposed  to  be 
ascertained  and  allowed  for,  but  not  the  variation ;  but 
in  the  true  course,  both  the  deviation  and  variation. 

The  leeway  is  the  oblique  motion  of  the  ship,  caused 
by  a  side  wind  driving  the  ship  along  a  track  oblique 
to  the  fore-and-aft  line,  and  therefore  not  indicated  by 
the  compass. 

The  amount  of  leeway,  under  a  wind  of  a  given 
obliquity  and  velocity,  for  each  ship  with  a  given 
freight,  is  best  found  by  trial. 

A  rhumb  line  is  the  track  of  a  ship  which  continues 
to  make  the  same  angle  with  the  meridians.  It  is  also 
called  a  loxodromic  curve. 


384  NAVIGATION. 

Since  the  meridians  converge,  the  rhumb  line  is  a 
spiral  curve. 

In  what  follows  we  shall  suppose  that  proper  allow- 
ances have  been  made  for  the  variation  and  deviation 
of  the  compass,  and,  therefore,  that  the  courses  given 
are  the  true  courses. 


383.    The  Log  and  Log  Line. 

The  log,  a  drawing  of  which  is  annexed,  is  a  board 
in  the  form  of  a  quadrant  whose  radius  is  about  six 
inches,  the  circular  part  of  which 
is  loaded  with  lead,  sufficient  to 
give  it  a  vertical  position  and  to 
cause  it  to  sink  so  that  the  vertex 
shall  be  just  above  the  surface. 

The  log  line  is  a  line  about  120  fathoms  in  length, 
and  so  attached  to  the  log  as  to  keep  its  face  toward 
the  ship,  that  it  may,  by  the  resistance  it  encounters 
from  the  water,  unwind  the  line  from  a  reel  as  the 
vessel  advances. 

The  log  line  is  divided  into  equal  parts  called  knots, 
each  knot  being  T^  of  a  nautical  mile,  or  50|  feet. 

The  time  is  measured  by  a  sand  glass,  through  which 
the  sand  passes  in  TJ¥  of  an  hour,  or  in  J  of  a  minute. 

Since  the  number  of  knots  in  a  nautical  mile  is 
equal  to  the  number  of  half-minutes  in  an  hour,  it 
follows  that  the  number  of  knots  run  off  in  half  a 
minute  is  equal  to  the  number  of  miles  the  ship  is 
sailing  an  hour. 

The  divisions  of  the  line  are  marked  by  strings  pass- 
ing through  the  line  and  knotted,  the  number  of  knots 
in  the  string  indicating  the  number  of  parts  between 


PLANE  SAILING. 


385 


it  and  that  point  of  the  line  where  the  divisions  com- 
mence at  that  end  of  the  line  next  to  the  log. 

The  stray  line  is  about  10  fathoms  of  the  end  of 
the  line  from  the  log  to  the  point  where  the  divisions 
begin.  This  portion  allows  the  log  to  settle  in  the 
water,  clear  of  th*e  ship,  before  the  measurement  of  the 
rate  begins. 

The  termination  of  the  stray  line  is  marked  by  a 
piece  of  red  cloth. 

The  sand  glass  is  turned  the  instant  this  cloth  passes 
the  reel,  which  is  stopped  the  moment  the  sand  has 
run  out. 

The  number  of  knots  on  the  string  which  marks 
the  last  division  run  from  the  reel,  indicates  the  rate 
of  sailing. 


PLANE   SAILING. 

384.    Single  Courses. 

Let  P  be  the  pole  of  the  earth  ;  RQ,  r 
the  equator;  AD,  a  rhumb  line  divided 
into  AB,  BC,  CD,  etc.,  parts  so  small  that 
we  may  regard  them  as  straight  lines; 
and  the  triangles  ABE,  BCF,  CDG,  plane 
triangles  and  similar,  which  give  the 
continued  proportions : 

AB  :  AE  : :  BC  :  BF  ::  CD  :  CG. 
AB  :  EB  : :  BC  :  FC  : :  .CD  :  GD. 

Since  the  sum  of  the  antecedents  is  to  the  sum  of 
the  consequents  as  one  antecedent  is  to  its  consequent, 
we  have, 

AD  :  AE+BF  +  CG  :  :  AB  :  AE. 

AD  :  EB  f  FC  +'GD  ::  AB  :  EB. 

S.  N.  33. 


386 


NAVIGATION. 


A d       B 


Now  let  a  right  triangle,  ABC,  be  con- 
structed, in  which  C  is  the  course  or  the 
angle  which  the  rhumb  line  makes  with 
the  meridian,  r  =  CB  =  AD,  the  rhumb 
line  of  the  first  figure;  I  --  CA  ==  AE  + 
HF  _|_  CG  =--  difference  of  latitude  ;  d  ^ 
AB  =  EB  +  FC  -\-  GD  ==  the  sum  of  the  elementary 
departures. 

We  may  now,  without  supposing  the  ship  to  sail  on* 
a  plane,  replace  the  surface  on  which  it  actually  sails 
by  a  plane  surface,  and  hence  the  name  plane  sailing. 


385.   Table  of  Cases. 


1 

Given. 

Req. 

Formulas. 

1 

r,  C, 

I,   d. 

I  ==  r  cos  (7,        d  —  r  sin  C. 

J 

2 

r,    I, 

C,d. 

cos  C=—  ,                d=i  >2-^. 
r 

j 

3 

A 

r,  d, 
ai 

C,l. 

v      fl 

•                 v-«                        1  &                                                              T                    ,     •'          O                    7  O 

sin  C=  —  ,                1  =  I  r2  —  a". 
r 

,7            /   fin    P 

*± 

5 

>  li 
C,d, 

r,  a. 
r,    I. 

cos  C 
d                            d 
~  sm~C'                "fanTc" 
^ 

6 

I,  d, 

r,  C. 

?•  =:l//2_|_^2^     fom  C=  y  • 

1.  —  /  .in    miles   may  be   reduced   to   degrees  by 
dividing  by  60. 

Note  2.  —  Examples  in  case  I.  may  be  solved   by  the 
Traverse  table. 


PLANE  SAILING.  387 

386.    Examples. 

1.  A  ship  sails  105  miles  N.  E.  by  N.,  from  latitude 
50° ;    required  the  latitude  in  which  the  ship  then  is, 
and  the  departure  made. 

Ans.  51°  27'.3  N.,  d  ==  58.34  mi. 

2.  A  ship  sailed  between  S.  and  W.  148  miles,  mak- 
ing the  difference  of  latitude  114.4;  required  the  course 
and  the  departure  made. 

Ans.  3i  pts.  W.  of  S.,  d  =  93.9  mi. 

3.  A  ship  in  latitude  3°  52'  S.  sails  between  N.  and 
W.  1065   miles,  making  a  departure  of  939  miles;    re- 
quired the  course  and  the  latitude  in  which  she  then  is. 

Ans.  N.  W.  b.  W.  JW.,  lat.  4°  30'  N. 

4.  A   ship   ran   from    latitude   38°  32'  N.  to  latitude 
36°  56'  N.  on   a   course  S.  E.  by  S.  JE. ;    required  the 
distance  sailed  and  the  departure  made. 

Ans.  r  ==  129.56  mi.,  d  =  87.009  mi. 

5.  A  ship  sailed  S.  56°  47'  E.  from  latitude  50°  13'  N. 
till  her  departure  was   82   miles;    required  r  and  lati- 
tude in.  Ans.  r  =  98  mi.,  lat.  49°  19'  N. 

6.  A  ship   from   latitude   36°  12'  N.  sails  between  S. 
and  W.  till  she  is  in  latitude  35°  1'  N.,  having  made 
76  miles  of  departure ;   required  r  and  C. 

Ans.  r  =±-  104  mi.,  C  =  S.  46°  57'  W. 

387.   Compound  Courses. 

A  compound  course  or  traverse  is  the  zigzag  course 
which  a  ship  usually,  takes  in  a  voyage  of  consider- 
able length. 

Working  the  traverse  is  the  computation  of  a  single 
course  and  distance  from  the  place  of  departure  to  the 
place  of  destination. 


388 


NAVIGATION. 


To  do  this,  find  by  the  Traverse .  table  the  latitude 
and  departure  of  each  course.  The  difference  of  the 
sum  of  the  northings  and  the  sum  of  the  southings 
will  be  the  latitude  of  the  single  course  required,  and 
the  difference  of  the  sum  of  the  eastings  and  the  sum 
of  the  westings  will  be  the  departure,  both  of  the  name 
of  the  greater.  Then  proceed  as  in  last  article. 

388.    Examples. 

1.  A  ship  sailed  from  latitude  51°  24'  N.  as  follows : 
S.  E.  40  miles,  N.  E.  28  miles,  S.  W.  by  W.  52  miles, 
N.  W.  by  W.  30  miles,  S.  S.  E.  36  miles,  S.  E.  by  E.  58 
miles;  required  the  latitude  in,  and  the  single  equiv- 
alent course  and  distance. 

Solution. 


Courses. 

Did.  \  AT.  L. 

S.  L. 

E.D. 

W.  D. 

S.  E. 

40 

28.3 

28.3 

N.  E. 

28 

19.8 

19.8 

43.2 

S.  W.  b.  W. 

52 

28.9 

24.9 

N.W.  b.  W. 

30 

16.7 

S.  S.  E. 

36 

33.3 

13.8 

S.  E.  b.  E. 

58 

32.2 

48.2 

36.5 

122.7 

110.1 

68.1 

36.5 

68.1 

86.2 

42 

42 


/  =  86.2  mi.  =  1°  26'. 


.  • .    C  =  25°  59'. 

!  ±=  95.87  mi. 

51°  24'— 1°  26'  — 49°  58'  N. 


2.  Given  the  following  courses  and  distances:    S.  W. 
W.  62  miles,  S.  by  W.  16  miles,  W.  J  S.  40  miles,  S.W. 


PARALLEL  SAILING. 

}W.  29  miles,  S.  by  E.  30  miles,    S.  f  E.   14  miles;    re- 
quired /,  (7,  and  r. 

Am.  1  =  1°  55'  S.,  (7=  S.  43°  14'  W.,  r  L  158  mi. 

3.  A  ship,  from  latitude  1°  12'  S.,  has  sailed  as  fol- 
lows: E.  by  N.  JN.  56  miles,  N.  }E.  80  miles,  S.  by  E. 
JE.  96  miles,  N.  JE.  68  miles,  E.  S.  E.  40  miles,  N.  N. 
W.  $}V.  86  miles,  E.  by  S.  65  miles;  required  the  lati- 
tude in,  C,  and  r. 

Ans.  Lat.  in,  0°  48'  N.,  C=  51°  47'  E.,  r  =  193.8  mi. 


PARALLEL   SAILING. 
389.   Definition  and  Principles. 

Parallel  sailing  is  that  case  of  sailing  in  which  the 
track  is  on  a  parallel  of  latitude. 

• 

Let  EFQ  be  the  equator; 

GAB,  the  parallel  of  the  track; 

r  —  AB  =  the  distance  sailed  ; 

L  —  FQ  =  the  difference  of  longitude  ; 

/  =  QB  —the  latitude  of  the  track. 

Since  similar  arcs  are  to  each  other  as  their  radii, 
(1)     DB  :  CQ  ::  AB  :  FQ. 

Consider  the  radius  CQ  as  the  unit  of  the  first  couplet, 
then  DB  will  be  the  natural  co-sine  of  latitude ;  and  take 
1  mile  as  the  unit  in  the  second  couplet,  put  r  for  AB, 
L  for  FQ,  then  (1)  becomes, 

(2)    COSl:l::r:L,      .:     (3)     L  =  -^  -. 

We  can  compute  L  in  (3)  by  taking  nat.  cos  Z,  or 
by  introducing  R  and  taking  log.  cos  I.  In  either  case 
L  will  be  found  in  miles,  since  r  is  given  in  miles; 
but  L  can  be  reduced  to  degrees  by  dividing  by  60. 


390  NAVIGATION. 

Let  r  and  r',  measured  on  the  parallels  whose  latitudes 
are  /  and  /',  respectively,  be  the  distances  between  two 
meridians  whose  difference  of  longitude  is  L. 

cos  I   :  1   : :  r   :  L, 


:  cos  /'  : :  r  :  /. 
cos  i  :  i  : :  r '  :  L,  ) 

Hence,  The  distances  between  two  meridians,  measured  on 
different  parallels,  are  as  the  co-sines  of  the  latitudes  of  tho$c 
parallels. 

To  find  the  length  of  a  degree  of  longitude  on  any 
parallel,  observe  that  at  the  equator  1°  of  Ion.  =  60 
nautical  miles,  and  that  cos  /  =  1,  then  we  shall  have, 

1  :  cos/'  ::  60  :  /,     .-.   /  =  60  cos  /'. 

390.   Examples. 

1.  A  ship  in -latitude  49°  32'  N.,  and  longitude  10° 
16'  W.,  sails  due  W.  118  miles;    required  the  longitude 
arrived  at.  Ans.  13°  18'  W. 

2.  A  ship  in  latitude   53°  36'  N.,  and  longitude   10° 
18'  E.,  sails  due  W.  236  miles;    required  the  longitude 
arrived  at.  Ans.  3°  40'  E. 

3.  A  ship  in  latitude  32°  N.  sails  6°  24'  due  W. ;    re- 
quired d.  Ans.  d  =  325.6  mi. 

4.  A  ship  sails  310  miles   from  longitude  81°  36'  W. 
to  longitude  91°  50'  W. ;    required  the  latitude  of  the 
track.  Ans.  59°  41'. 

MIDDLE   LATITUDE   SAILING. 
391.   Definition  and  Principles. 

Middle  latitude  sailing  is  a  combination  of  plane  sail- 
ing and  parallel  sailing,  on  the  supposition  that  the 
departure  in  plane  sailing  is  equal  to  the  distance 


MIDDLE  LATITUDE  SAILING.  391 

between  the  meridians  passing  through  the  extreme 
points  of  the  rhumb  line,  measured  on  the  middle 
parallel  between  these  points. 

Let  AD  be  a  rhumb  line;  IK,  the 
middle  parallel ;  w,  the  latitude  of  IK; 
then  d  =  EB  +  FC  +  GD  =  IK. 

For .  r,  formula  (3),  parallel  sailing, 
,substitude  d  or  its  value  as  found  in 
plane  sailing;  and  for  cos  I  substitute 
cos  ?»,  then  we  shall  have, 


T  d        -  r  sin  C       l'>2  — J2        I  tan  C 

cos  m         cos  m  cos  m  cos  m 

Note  1.  —  Remember  that  in  these  formulas  I  denotes 
the  difference  of  latitude ;  L ,  the  difference  of  longitude 
in  miles;  d,  the  departure;  r,  the  distance  run  or  the 
rhumb  line;  C,  the  course,  and  m,  the  middle  latitude. 

Note  2.  —  The  middle  latitude  is  the  half  sum  of  the 
extreme  latitudes ;  or  the  less  latitude,  plus  the  half 
difference  of  latitude ;  or  the  greater  latitude,  minus 
the  half  difference  of  latitude. 

Note  3. — That  the  departure  is  not  strictly  equal  to  the 
middle-latitude  distance  between  the  meridians,  through 
the  extremities  of  the  rhumb  line,  is  thus  shown  : 

Suppose  a  ship  to  sail  on  this  middle  latitude  from 
one  of  the  meridians  to  the  other,  then  the  distance 
{sailed  wrill  be  the  departure;  but  if  a  second  ship  were 
jjto  sail  from  a  lower  latitude  on  the  first  meridian, 
imd  a  third  ship,  from  a  higher,  to  the  same  place, 
Ithe  departure  of  the  second  would  be  greater,  and  the 
departure  of  the  third  woulcfc  be  less  than  that  of  the 
first. 

It  is  necessary,  therefore,  to  make  the  correction  for 
middle  latitude  as  found  in  the  table  for  such  corrections. 


392  NAVIGATION. 

The  following  is  the  rule  for  correcting  the  middle 
latitude  : 

Add  to  the  unconnected  middle  latitude  the  correc- 
tion found  in  the  table  under  the  difference  of  latitude, 
and  opposite  the  middle  latitude  —  the  sum  m'  is  the 
corrected  middle  latitude. 

7-  -        d         _  rsin  <1     _  i  V2ZT^2  _     /  tan  C 
cos  m'         cos  m'  cos  mf  cos  mf 

392.    Examples. 

1.  A  ship  from  latitude  51°  18'  N.,  longitude  9°  50' 
W.,  sails  S.  33°  8'  W.  1024  miles ;    required  the  latitude 
and  longitude  in. 

I  ------  r  cos  C,     .'.  1  =  857.4  mi.  M  14°  17'. 

•  .-.     51°  18'—  14°  17=  37°  1',  the  hit.  in. 
4(51°  18'+ 37°  I')' =  44°  9J'=mid.  lat.,  correction  =  27'. 
44°  9J'  -f-  27'  =  44°  36J'  =  m'  =  corrected  mid.  lat. 

y  m  11   \ 

L  =  -      ^  ,     .-.  L  =  786.3  mi.  ==  13°  6'. 
cos  m 

9°  50'  +  13°  6'  --=  22°  56'  W.,  the  Ion.  in. 

2.  A  ship,  from  latitude  52°  6'  N.,  and  longitude  35° 
6'  W.,  sails  N.  W.  by  W.  229  miles;    required  the  lati- 
tude and  longitude  arrived  at. 

Am.  Lat.  54°  13'  N.  and  Ion.  40°  23'  W. 

3.  A  ship  from  latitude  49°  57'  N.,  and  longitude  5° 
11'  W.,  sails  between  S.  and  W.  till  she  is  in  latitude 
38°  27'  N.,  when   she   has   made   440  miles  departure; 
required  (7,  r,  and  the  longitude  in. 

An8.  C  =  S.  32°  32'  W. ;  r  &±  818  mi. ;  Ion.  in,  15°  28'  W. 

4.  A  ship  from  latitude  37°  N.,  longitude  22°  56'  W., 
sails   N.  33°   19'   E.  till   she   is   in   latitude  51°  18'  N. 
What  longitude  is  she  in?-  Ans.  9°  45'  W. 


MERCATOR'S  SAILING,  393 

5.  A  ship  from  latitude  40°  41'  N.,  longitude  16°  37' 
W.,  sails  between  N.  and  E.  till  she  is  in  latitude  43° 
57'  N.,  and  finds  that  she  has  made  248  miles  departure ; 
required  (7,  r,  and  longitude  in. 

Ans,  C=  51°  41'  E. ;   r  =  316  mi.;  Ion.  in,  11°  W. 


MERCATOR'S  SAILING. 

393.   Definitions  and  Principles. 

Mercator's  chart,  so  called  from  its  originator,  Gerrard 
Mercator,  a  Fleming,  who  first  published  it  in  1556,  is 
a  representation  of  the  surface  of  the  earth  on  the  sup- 
position that  the  earth  is  a  cylinder. 

The  meridians  are  thus  represented  parallel  and 
every-where  too  far  apart  except  at  the  equator. 

To  guard  as  much  as  possible  against  distortion,  the 
distances  between  the  parallels  are  proportionally  in- 
creased. 

The  surface  is  thus  relatively  magnified  more  and 
more  toward  the  poles. 

Mercator's  sailing  is  the  method  of  computing  the 
difference  of  longitude  from  the  principle  on  which 
Mercator's  chart  is  projected. 

The  mathematical  theory  of  this  method  was  devel- 
oped, and  the  Table  of  Meridional  Parts,  necessary  to  its 
application,  computed  by  Edward  Wright,  an  English- 
man, in  1599.  A' 

Let  CA  and  AB.  respectively,  be  the  dif- 
ference  of  latitude  and  departure  corre- 
sponding to  the  rhumb  line  (75,  and  let 
CA  be  produced  to  A'  till  A'B',  the  corre- 
sponding departure,  is  equal  to  the  differ-  c 


394 


NAVIGATION. 


ence  of  longitude  of  C  and  B.  CA'  is  called  the  'merid- 
ional difference  of  latitude,  which  is  simply  the  proper 
difference  of  latitude  increased  till  the  corresponding 
departure  is  equal  to  the  difference  of  longitude  corre- 
sponding to  the  proper  departure. 

To  find  the  meridional  difference  of  lati- 
tude, let  Cb,  bd,  df,  ...  be  indefinitely  small 
portions  of  the  rhumb  line  CB.  Ca,  be,  de, 
. . .  corresponding  differences  of  latitude ; 
ab,  cd,  ef,...  corresponding  differences  of 
departure ;  Ca',  be',  de',  .  . .  corresponding 
meridional  differences  of  latitude ;  a'b',  c'd', 
ef,  . . .  differences  of  longitude  corresponding  to  the 
departures  ab,  cd,  ef,  . . .  whose  latitudes  are  /,  I',  /",  . . . 
Then,  as  found  in  Parallel  sailing, 


ab  :  a'b'  : :  cos  I  :  1. 
but  ab  :  a'b'  : :  Ca  :  Ca'. 
• .  cos  I  :  1  :  :  On  :  Ca',  .  • .  Ca'  = 


but        — 7  =  sec  /, 
cos  / 

In  like  manner. 


Ca 


cos  I 
.  Ca'  =  Ca  sec  /. 

6^  =  be  sec  /', 
dJ  =  rfg  sec  r. 


But  CA'  =  Ca'  -1-  be'  +  de'  +  ... 

Substituting  the  values  of  Ca',  be',  de',  . . .  and  making 

Ca .  =  be  =  de  =  . . .  =  V,  we  have, 

CA  =  sec  I  -j-  sec  I'  -f  sec  l"-\-  .  . . 

Commencing   at   the   equator,  and   putting   m.  p.  for 
meridional  parts,  and  taking  natural  secants,  we -have, 

m.  p.  of  r  =  sec  1'. 

m.  p.  of  2'  =  sec  1'  -(-  sec  2'. 


MERC  A  TOR'S  SAILING.  395 

w.  p.  of  3'  —  sec  V  -f  sec  2'  -{-  sec  3'. 

m.  p.  of  4'  ==  sec  1'  -)-  sec  2'  -f-  sec  3'  -f-  sec  4'. 

By  substituting  and  condensing,  we  have, 

m.p.  of  1'  =  1.0000000  =  1.0000000 

m.y,  of  2'  4:  1.0000000  +  1.0000002  s=  2.0000002 
m.  p.  of  3'  =  2.0000002  -f  1.0000004  =  3.0000006 
m.p.  of  4'=  3.0000006  +  1.0000007  =  4.0000013 

The  accuracy  of  the  result  is  increased  by  taking  the 
parts  still  smaller,  as  %. 

Having  found  the  meridional  latitude  corresponding 
to  C,  and  also  to  J,  their  difference  will  be  the  merid- 
ional difference  of  latitude  found  from  the  table;  and 
the  corresponding  departure,  A'B',  will  be  the  differ- 
ence of  longitude. 

Denoting  the  proper  difference  of  latitude  CA  by  /, 
the  meridional  difference  of  latitude  by  /',  the  departure 
AB  by  d,  and  the  difference  of  longitude  A'B'  by  L,  the 
triangles  CAB  and  CAB'  give, 

1  :  tan  C  : :  /'  :  -L,     .  • .   L  =  I'  tan  C. 
I   :          d  ::  V  :  L,     .-.  L=™- 

394.   Examples  in  Single  Courses. 

1.  A  ship  from  latitude  52°  6'  N.,  and  longitude  35° 
&  W.,  sails  N.  W.  by  W.  229  miles;  required  the  lati- 
tude and  longitude  in. 

I  =  r  cos  C  =  229  cos  56°  15',     .',  1=  127.3  mi.-=  2°  7'. 
lat.  in  ==  52°  6'  N.  +  2°  7'  N.  =  54°  13'  N. 


396  NAVIGATION. 


m.p.  of  54°  13'  ==  3868 
w..of52°    6'  =  3657 


But  L  =  I'  tan  <7, 

.  •  .    L  =  211  tan  56°  15'. 


or     L  =  315.8  mi.  =  5°  16'. 
,  • .  Ion.  in  -  35°  6'  W.  -f  5°  16'  W.  =  40°  22'  W. 

2.  A  ship  from  latitude  51°  18'  N.,  and  longitude  9° 
50'  W.,  sails  S.  33°  8'  W.  1024  miles ;  required  the  lati- 
tude and  longitude  in. 

Ans.  Lat.  in  37°  1'  N. ;  Ion.  in  22°  50'  W. 

3.  Required    the   course  and   distance  from   Ushant, 
latitude  48°  28'  N.,  longitude  5°  3'  W.,  to  St.  Michael's, 
latitude  37°  44'  N.,  longitude  25°  40'  W. 

Ans.  S.  54°  3(X  W.,  r  =  1106  mi. 

4.  A  ship  from  latitude  51°  9'  N.  sails  S.  W.  b.  W. 
216  miles;  required  the  latitude  in,  and  the  difference 
of  longitude  made.         Ans.  Lat.  49°  9'  N.,  1=4°  39'. 

5.  A  ship  sails  from  latitude  37°  N.,  longitude  22° 
56'  W.,  on  the  course  N.  33°  19'  E.,  till  she  arrives  at 
latitude  51°   18'   N. ;    required  the  distance  sailed  and 
the  longitude  arrived  at.     Ans.  1027  mi.,  Ion.  9°  47'  W. 

6.  A  ship  sails  N.  E.  b.  E.  from  latitude  42°  25'  N., 
and  longitude  15°  6'  W.,  till  she  finds  herself  in  latitude 
46°  20'  N. ;  required  the  distance  sailed  and  the  longi- 
tude in.  Ans.  Dist.,  423  mi. ;  Ion.  6°  55'  W. 

395.   Examples  in  Compound  Courses. 

1.  A  ship  from  latitude  60°  9'  N.,  and  longitude  1° 
7'  W.,  sailed  as  follows:  N.  E.  b.  N.,  69  miles;  N.  N.  E., 
48  miles;  N.  b.  W.  £W.,  78  miles;  N.  E.,  108  miles; 
S.  E.  b.  E.,  50  miles;  required  the  latitude  and  longi- 
tude in,  and  the  direct  course  and  distance. 


MERCA  TOP'S  SAILING. 


397 


Courses. 

Did. 

JV.  L. 

-v.  /,. 

Lot. 

m.  p. 

m.d.l. 

E.L. 

W.L. 

N.  E.  b.  N. 

69 

57.4 

60°9X 

4525 

N.  N.  E. 

48 

44.4 

6106X 

4641 

116 

77.5 

N.b.W.^W. 

78 

74.6 

WtW 

4733 

92 

38.1 

N.  E. 

108 

76.4 

63°5/ 

4895 

162 

49. 

S.  E.  b.  E. 

50 

27.8 

64°21/ 

5067 

172 

172.0 

' 

252.8 

63°53X 

5003 

64 

95.8 

27.8 

Dif.  hit.  =  I  =  225  mi.  =  3°  45'  N. 


383.4 


Lat.  Left  =  60°  9'  N. 
Dif.  Lat.  =  3°  45'  N. 
Lat.  in  =s  63°  54'  N. 


Dif.  Ion.  =  L  ==  334.4  mi.  == 
Dif.  Ion.  =  5°  34'  E. 
Lon.  left  =  1°   7'  W. 
Lon.  in  :^4°~27TET 


m.  p.  of  lat.  in  (63°  540  :-  5005. 
m.  p.  of  lat.  left  (60°  9')  ±=  4525. 
Meridional  dif.  lat.  =1'  =  --  '  480. 


n       L        334.4 
tan  C  .3=  r  — 

r         480 


I 


225 


cos  C       cos  34°  53' 


=  N.  34°  53'  E. 

.  • .  r  =  273  mi. 


2.  A  ship  from  latitude  38°  14'  N.,  and  longitude  25° 
56'  W.,  has  sailed  the  following  courses  :  N.  E.  b.  N.  JE., 
56  miles;  N.  N.  W.,  38  miles;  N.  W.  b.  W.,  46  miles; 
S.  S.  E.,  30  miles;  S.  b.  W.,  20  miles;  N.  E.  b.  N.,  60 
miles;  required  the  latitude  and  longitude  in,  and  the 
direct  single  course  and  distance. 

Ana    Lat.   in,    40°    2'.  3  N. ;    Ion.   in,  25.°   30'  W. ; 
(7=  N.  10°  33'  E.,  r  =i  110.2  mi. 

390.   Correction  for  Middle  Latitude. 

We  are  now  prepared  to  understand  how  the  correc- 
tion for  middle  latitude,  before  used,  is  found. 


NAVIGATION. 

I  denotes  the  proper  difference  of  latitude ; 

I',  the  meridional  difference  of  latitude ; 

.L,  the  difference  of  longitude ; 

m,  the  middle  latitude  uncorrected; 

c,  the  correction; 

?>i',  the  middle  latitude  corrested. 

Then,  by  Plane,  Middle  latitude,  and  Mercator's  sailing, 

d         L  cos  m'         L  I 

tan  C  =  -j-  =  -, =  —p-  ,     . ' .  cos  m  =  y 

From  which  m'  is  readily  found. 
Then,  c  =  in'  —  m.     .  • .  m'  =  m  -(-  c. 


CURRENT   SAILING. 
397.    Definition  and  Principles. 

Current  sailing  is  the  sailing  of  a  ship  as  affected 
by  a  current. 

Irrespective  of  the  current  the  ship  would  move,  in 
a  certain  time,  a  certain  course  and  distance. 

The  current  alone  would  carry  the  ship,  in  the  same 
time,  a  certain  other  course  and  distance. 

The  actual  track  of  the  ship,  which  is  the  resultant 
of  the  two,  will  bring  her  to  the  same  position  as  if 
she  had  sailed  separately  the  two  tracks. 

Current  sailing  may  therefore  be  treated  as  Plane 
sailing,  compound  courses. 

The  set  of  the  current  is  its  direction. 
The  drift  of  the  current  is  its  velocity. 

The  set  and  drift  of  a  current  may  be  ascertained 
by  taking,  a  short  distance  from  the  ship,  a  boat,  which 
is  kept  from  being  carried  by  the  current  by  letting 


CURRENT  SAILING. 


399 


down,  to  a   considerable  depth,  a  heavy  weight,  which 
is  attached  by  a  rope  to  the  stern  of  the  boat. 

The  log  being  thrown  from  the  boat  into  the  current, 
the  direction  in  which  it  is  carried,  or  set  of  the  cur- 
rent, is  determined  by  the  boat  compass,  and  the  rate 
at  which  it  is  carried,  or  drift  of  the  current,  by  the 
number  of  knots  of  the  log  line  run  out  in  half  a 
minute. 

398.   Examples. 

1.  A  ship  sails  N.  W.  a  distance,  by  the  log,  of  60 
miles,  in  a  current  that  sets  S.  S.W.,  drifting  25  miles 
in  the  same  time;  required  the  course  and  distance. 


Courses. 

Dist. 

N.L. 

8.'L 

KD. 

W.D. 

N.  W. 

60 

42.4 

42.4 

S.  S.  W. 

25 

23.1 

9.6 

I  =  19.3. 


d  =  52. 


—  "^Jff..,    ...  c=  N.  69°  38'  W. 


r  = 


=  V  (19.3)2+  (52)  2  ==  55.5. 


2.  A  ship,  sailing  7  knots  an  hour,  is  bound  to  a  port 
bearing  S.  52°  W.,  through  a  current  S.  S.  E.,  2  miles 
an  hour;  required  the  course.  N 

Let  AB  be  the  direction  of  the 
port. 

AE,  the  direction  of  the  current, 
_  o 

AD,  the  required  direction,  =  7. 
Complete  the  parallelogram,  DBA 
=  BAE  =  52°  +  22°  3(X  =  74°  30'.    Then  we  have, 


400  NAVIGATION. 

AD  :  DE  :  :  sin  DBA  :  sin  DAB. 

2  sin  DBA 
-    — —  —  • 

.'.    (7—15°  59' +  52°  =  67°  59'. 

3.  A  ship  runs  N.  E.  by  N.  18  miles  in  3  hours,  in  a 
current  W.  by  S.  2  miles  an  hour;    required  the  course 
and  distance.  Ans.  C  =  N.  b.  E.  JE.,  r  ===  14  mi. 

4.  In  a  current  S.  E.  by  S.  1J  miles  an  hour,  a  ship 
sails  24  hours  as  follows:   S.W.,  40  miles;  W.  S.W.,  27 
miles  ;  S.  by  E.,  47  miles;  required  the  direct  course  and 
the  distance.         Ans.  C  =  S.  11°  50'  W.,  r  =  117  mi. 

5.  The  port  bears  due  E.,  the  current  sets  S.  W.  by 
S.  3   knots   an   hour,  the  rate  of  sailing  is  4  knots  an 
hour;   required  the -course  steered.         Ans.  N.  51°  E. 

6.  A  ship  sailing  in  a  current  has,  by  her  reckoning, 
run  S.  by  E.  42  miles,  and,  by  observations,  is  found  to 
have  made   55   miles  of  difference   of  latitude,  and   18 
miles  of  departure;    required   the  set  and  drift  of  the 
current.       Ana.  Set,  S.  62°  12'  W. ;  whole  drift,  30  mi. 


PLYING   TO  WINDWARD. 
399.    Definitions. 

Plying  to  windward  is  the  zigzag  course  which  a  ship 
makes  by  tacking  when  she  encounters  a  foul  wind. 

Starboard  signifies  the  right  side. 

Larboard  signifies  the  left  side. 

The  starboard  tacks  are  aboard  when  a  ship  plies 
with  the  wind  on  the  right. 

The  larboard  tacks  are  aboard  when  a  ship  plies 
with  the  wind  on  the  left. 


PLYING  TO  WINDWARD.  401 

A  ship  i^  said  to  be  dose-hauled  when  she  sails  as 
nearly  as  poppibJ.e  toward  the  point  from  which  the 
wind  is  blowing. 

400.    Examples. 

1.  Being  within  sight  of  my  port   bearing    N.  by  E. 
^E.,  distant  18  miles,  a  fresh  gale  sprung  up  from  the 
N.  E.    With  my  larboard  tacks  aboard,  and  close-hauled 
within  six  points  of  the  wind,  how  far  must  I  run  be- 
fore tacking  about,  and  what  will  be  my  distance  from 
the  port  on  the  second  board  ?  N 

Let  A  be  the  place  of  the  ship ; 
P,  the  port;  ABy  the  distance  of 
the  first  board;  BP,  that  of  the 
second;  WA  or  W'By  the  direction 
of  the  wind. 

Then,  WA  B  '•=-.  W'BC  =  W'BP  = 
6  points. 

.  • .    ABP  --16  points  —  12  points  =  4  points. 

PA  W-=  NA  W—  NAP  =  4  points  —  \\  points  =3  2£ 

points. 
PAB  =  PAW -f-  WAB  =  24  points  -f  6  points  ==  8J 

points. 

APB  ==  16  points  —  (PAB  +  ABP)  =  3J  points, 
sin  ABP  :  sin  APB  :  :  AP  :  AB,       .  •.  AB  =  16.15  mi. 
sin  ABP  :  sin  BAP  :  :  AP  :  BP,        .  *.    BP  =  25.23  mi. 

2.  If  a  ship  can  lie  within  6  points  of  the  wind  on 
the   larboard   tack,  and  within   5J   points   on  the  star- 
board  tack;    required  her  course  and  distance  on  each 
tack  to  reach  a  port  lying  S.  by  E.  22  miles,  the  wind 

being  at  S.  W. 

(  Starboard  tack,  S.  b.  E.  JE.  23.66  mi. 

-  Ans'  \  Larboard  tack,  W.  N.  W.~2.79  mi. 
S.  N.  34. 


402  NAVIGATION. 

3.  A  ship   is  bound    to   a   port  80  miles  distant,  and 
directly   to  windward,  which    is    N.  E.  by  N.  -J^.,  and 
proposes  to  reach  her  port   at  two  boards,  each  within 
6  points  of  the  wind,  and  to  lead  with   the   starboard 
tack ;    required  her  course   and   distance  on  each  tack. 

(  Starboard  tack,  X.  N.W.  iW.,  104.5  mi. 
Am'    \  Larboard  tack,  E.  S.  E.  iE.,  104.5  mi. 

4.  Wishing   to    reach    a    point    bearing    N.   N.  W.   15 
miles,  but   the  wind  being  at  W.  by  N.,  I  was  obliged 
to  ply  to  windward — the  ship,  close-hauled,  could  make 
way  within  6  points  of  the  wind;    required  the  course 
and  distance  on  each  tack. 

f  Larboard  tack,  N.  b.  W.  17.65  mi. 
Ans'   I  Starboard  tack,  S.W.  b.  S.  4.138  mi. 


TAKING    DEPARTURES. 
401.    Explanation. 

Before  losing  sight  of  land,  at  the  beginning  of  %a 
voyage,  the  bearing  and  distance  of  some  well-known 
object,  as  a  light-house  or  headland,  is  taken,  the  re- 
verse bearing  and  distance  of  which  are  entered  as 
the  first  course  and  distance  on  the  log  board. 

The.  bearing  is  taken  by  the  compass;  but  the  dis- 
tance is  sometimes  estimated  by  the  eye,  as  can  be 
done  with  considerable  accuracy  by  navigators  of  ex- 
perience. 

A  more  correct  method  of  taking  a  departure  is  by 
means  of  data,  obtained  by  taking  the  bearing  at  two 
different  positions  of  the  ship,  the  distance  between 
these  positions  being  measured  by  the  log. 


TAKING   DEPARTURES.  403 

402.    Examples. 

1.  Sailing  down  the  channel,  the  Eddystone  bore  N.W. 
by  N.,  and  after  running  W.  S.  W.  18  miles,  it  bore  N. 
by  E.;   required  the  course  and  distance  from  the  Eddy- 
stone    to   the    place   of   the    last   obser-  E 
vation. 

E  =  NAE  -f  N'BE  =  4  points. 
A  =  16  points  —  (NAE  +  BAS)  =  7  pts. 
sin  E  :  sin  A  :  :  AB  :  BE, 
.  • .     BE=  24.97. 

2.  At  3  o'clock  P.  M.  the  Lizard  bore  N.  by  W.  £ 
and  after  sailing  7  knots  an  hour,  W.  by  N.  JN.,  till  6 
o'clock,  the  Lizard  bore  N.  E.  f  E. ;    required  the  course 
and  distance  from   the   Lizard  to  the  place  of  the  last 
observation.  Ans.  S.W.  |W.,  19.35  mi. 

3.  In  order  to  get  a  departure,  I  observe  a  headland 
of  known  latitude  and  longitude  to  bear  N.  E.  by  N., 
and  after  sailing  E.  by  N.  15  miles,  the  same  headland 
bore  W.  N.W. ;  required  my  distance  from  the  headland 
at  each  place  of  observation. 

Ans.  8.5  mi.  and  10.8  mi. 

Remark. — To  find  the  latitude  and  longitude  of  a  ship 
by  means  of  celestial  observations,  requires  a  knowledge 
of  Nautical  Astronomy;  but  a  thorough  discussion  of 
this  subject  would  require  an  amount  of  space  far  ex- 
ceeding our  limits. 


TABLES. 


I.  LOGARITHMS  OF  NUMBERS, 
II.  NATURAL  SINES  AND  CO-SINES, 

III.  NATURAL  TANGENTS  AND  CO-TANGENTS, 

IV.  LOGARITHMIC  SINES  AND  TANGENTS, 

V.  TRAVERSE  TABLE, 

VI.  MISCELLANEOUS  TABLE,    . 
VII.  MERIDIONAL  PAHTS,          . 
VIII.  CORRECTIONS  TO  MIDDLE  LATITUDES,     . 


PAOK. 
.         1 

.     24 

.     26 

.     28 
.     73 

.     84 

.     85 

87 


Logarithms  of  Numbers  to  100. 


1 

0  00000 

21 

1.32222 

41 

161278 

61 

1.78533 

81 

1.90849 

2 

0.30103 

22 

1.34242 

42 

1.62325 

62 

1  79239 

82 

1.91381 

3 

047712 

23 

.36173 

43 

1.63347 

63 

1.79934 

83 

1.91908 

4 

0.60206 

24 

.38021 

44 

.64345 

64 

1.80618 

84 

1.92428 

a 

0.69897 

25 

.39794 

45 

.65321 

65 

1.81291 

85 

1.92942 

6 

0.77815 

26 

.41497 

46 

.66276 

66 

1.81954 

86 

1.93450 

7 

0.84510 

27 

.43136 

47 

.67210 

67 

1.82607 

87 

.93952 

8 

0.90309 

28 

.44716 

48 

.68124 

68 

1.83251 

88 

.94448 

9 

0.95424 

29 

.46240 

49 

.69020 

69 

1  83885 

89 

.94939 

10 

.00000 

30 

.47712 

50 

.69897 

70 

1.84510 

90 

.95424 

11 

1.04139 

31 

.49136 

51 

.70757 

71 

-1.85126 

91 

.95904 

12 

.07918 

32 

.50515 

52 

71600 

72 

1.85733 

92 

1.96379 

13 

.11394 

33 

.51851 

53 

.72428 

73 

1.86332 

93 

.96848 

14 

.14613 

34 

.53148 

54 

.73239 

74 

1.86923 

94 

.97313 

15 

.17609 

35 

,54407 

55 

.74036 

75 

1.87506 

95 

.97772 

16 

.20412 

36> 

.55630 

56 

.74819 

76 

1.88081 

96 

.98227 

17 

.23045 

37 

.56820 

57 

.75587 

77 

1.88649 

97 

.98677 

18 

.25527 

38 

.57978 

58 

.76343 

78 

1.89209 

98 

.99123 

19 

.27875 

39 

.59106 

59 

.77085 

79 

1.89763 

99 

1.99564 

20 

.30103 

40 

1.60206 

60 

.77815 

80 

1.90309 

100 

2.00000 

(1) 


100-144 


LOGARITHMS. 


00000-16107 


I, 

0 

1 

2 

3 

4 

5  !  '6    78    9 

D. 

100 

00000 

043 

087 

130 

173 

217 

260 

303 

346 

389 

43 

101 

432 

475 

518 

561 

604 

647 

689 

732 

775 

817 

43 

102 

860 

903 

945 

988 

o30 

o72 

i!5 

i57 

i99 

242 

42 

103 

01284 

326 

368 

410 

452 

494 

536 

578 

620 

662 

42 

104 

703 

745 

787 

828 

870 

912 

953 

995 

o36 

o78 

42 

105 

02119 

160 

202 

243 

284 

325 

366 

407 

449 

490 

41 

106 

531 

572 

612 

653 

694 

735 

776 

816 

857 

898 

41 

107 

938 

979 

o!9 

o60 

i()0 

i41 

18! 

222 

262 

302 

40 

108 

03342 

383 

423 

463 

503 

54:) 

583 

623 

663 

703 

40 

109 

743  782 

822 

862 

902 

941 

981 

o21 

o60 

lOO 

40 

110 

04139 

179 

218 

258 

297 

336  i  376 

415 

454 

493 

39 

111 

532 

571 

610 

650 

689 

727 

766 

805 

844 

883 

39 

112 

922 

961 

999 

o38 

o77 

.i!5 

1-34 

i92 

231 

269 

39 

113 

05308 

346 

385 

423 

461 

500 

538 

576 

614 

652 

38 

114 

690 

729 

767 

805 

843 

881 

918 

956 

994 

o32 

38 

115 

06070 

108 

145 

183 

221 

258 

296 

333 

371 

408 

38 

116 

446 

483 

521 

558 

595 

633 

670 

707 

744 

781 

37 

117 

819 

856 

893 

930 

967 

o()4 

o41 

o78 

i!5 

iol 

37 

118 

07188 

225 

262 

298 

335 

372 

408 

445 

482 

518 

37 

119 

555 

591 

628 

664 

700 

737 

773 

809 

846 

882 

36 

120 

918 

954 

990 

o27 

o63 

o99 

i35 

i71 

207 

243 

36 

121 

08279 

314 

350 

386 

422 

458 

493 

529 

565 

600 

36 

122 

636 

672 

707 

743 

778 

814 

849 

884 

920 

955 

35 

123 

991 

o26 

06  1 

o96 

1  32 

i67 

202 

237 

272 

s07 

35 

124 

09342 

377 

412 

447 

482 

517 

552 

587 

621 

656 

35 

125 

691 

726 

760 

795 

830 

864 

899 

934 

968 

o03 

35 

126 

10037 

072 

106 

140 

175 

209 

243 

278 

312 

346 

34 

127 

380 

415 

449 

483 

517 

551 

585 

619 

653 

687 

34 

128 

721 

755 

789 

823 

857 

890 

924 

958 

992 

o25 

34 

129 

11059 

093 

126 

160 

193 

227 

261 

294 

327 

361 

34 

130 

394 

428 

461 

494 

528 

561 

594 

628 

661 

694 

33 

131 

727 

760 

793 

826 

860 

893 

926 

959 

992 

o24 

33 

132 

12057 

090 

123 

156 

189 

222 

254 

287 

320 

352 

33 

133 

385 

418 

450 

483 

516 

548 

581 

613 

646 

678 

33 

134 

710 

743 

775 

808 

840 

872 

905 

937 

969 

oOl 

32 

135 

13033 

066 

098 

130 

162 

194 

226 

258 

290 

322 

32 

136 

354 

386 

418 

450 

481 

513 

545 

577 

609 

640 

32 

137 

672 

704 

735 

767 

799 

830 

862 

893 

92f) 

956 

32 

138 

988 

o!9 

o51 

o82 

i!4 

i45 

i76 

208 

239 

270 

31 

139 

14301 

333 

364 

395 

426 

457 

489 

520 

551 

582 

31 

140 

613 

644 

675 

706 

737 

768 

799 

829 

860 

891 

31 

141 

922 

953 

983 

o!4 

o45 

(.76 

i06 

i37 

168- 

i98 

31 

142 

15229 

259 

290 

320 

351 

381 

412 

442 

473 

503 

31 

143 

534 

564 

594 

625 

655 

685 

715 

746 

776 

806 

30 

144 

836 

866 

897 

927 

957 

987 

o!7 

o47 

o77 

i07 

30 

y, 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

145-189 


LOGARITHMS. 


16137-27852 


I, 

0 

1 

2 

3  ' 

4 

5 

6 

7 

8 

9 

D, 

145 

16137 

167 

197 

227 

256 

286 

316 

346 

376 

406 

30 

146 

435 

465 

495 

524 

554 

584 

613 

643 

673 

702 

30 

147 

732 

76.1 

791 

820 

850 

879v 

909 

938 

967 

997 

29 

148 

17026 

056 

085 

n4 

143 

173 

202 

231 

260 

289 

29 

149 

319 

348 

377 

406 

435 

464 

493 

522 

551 

580 

29 

150  • 

609 

638 

667 

696 

725 

754 

782 

811 

840 

869 

29 

51 

898 

926 

955 

984 

o!3 

o41 

o70 

o99 

i27 

i56 

29 

152 

18184 

213 

241 

270 

298 

327 

355 

384 

412 

441 

29 

1  53 

469 

498 

526 

554 

583 

611 

639 

667 

696 

724 

28 

154 

752 

780 

808 

837 

865 

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921 

949 

977 

o05 

28 

155 

19033 

061 

089 

117 

145 

173 

201 

229 

257 

285 

28 

156 

312 

340 

368 

396 

424 

451 

479 

507 

535 

562 

28 

157 

590 

618 

645 

673 

700 

728 

756 

783 

811 

838 

28 

158 

866 

893 

921 

948 

976 

o03 

o30 

o58 

o85 

i!2 

27 

159 

20140 

167 

194 

222 

249 

276 

303 

330 

358 

385 

27 

160 

412 

439 

466 

493 

520 

548 

575 

602 

629 

656 

27 

161 

683 

710 

737 

763 

790 

817 

844 

871 

898 

925 

27 

162 

952 

978 

o05 

o32 

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i!2 

i39 

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27 

163 

21219 

245 

272 

299 

325 

352 

378 

405 

431 

458 

27 

164 

484 

511 

537 

564 

590 

617 

643 

669 

696 

722 

26 

165 

748 

775 

801 

827 

854 

880 

906 

932 

958 

985 

26 

166 

22011 

037 

063 

089 

115 

141 

167 

194 

220 

246 

26 

167 

272 

298 

324 

350 

376 

401 

427 

453 

479 

505 

26 

168 

531 

557 

583 

608 

634 

660 

686 

712 

737 

763 

26 

169 

789 

814 

840 

866 

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917 

943 

968 

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o!9 

26 

170 

23045 

070 

096 

121 

147 

172 

198 

223 

249 

274 

25 

171 

300 

325 

350 

376 

401 

426 

452 

477 

502 

528 

25 

72 

553 

578 

603 

629 

654 

679 

704 

729 

754 

7T9 

25 

73 

805 

830 

855 

880 

905 

930 

955 

980 

o05 

o30 

25 

174 

24055 

080 

105 

130 

155 

180 

204 

229 

254 

279 

25 

75 

304 

329 

353 

378 

403 

428 

452 

477 

502 

527 

25 

76 

551 

576 

601 

625 

650 

674 

699 

724 

748 

773 

25 

177 

797 

822 

846 

871 

895 

920 

944 

969 

993 

o!8 

25 

178 

25042 

066 

091 

115 

139 

164 

188 

212 

237 

261 

24 

179 

285 

310 

334 

358 

382 

406 

431 

455 

479 

503 

24 

180 

527 

551 

575 

600 

624 

648 

672 

696 

720 

744 

24 

181 

768 

792 

816 

840 

864 

888 

912 

935 

959 

983 

24 

182 

26007 

031 

055 

079 

102 

126 

150 

174 

198 

221 

24 

183 

245 

269 

293 

316 

340 

364 

387 

411 

435 

458 

24 

184 

482 

505 

529 

553 

576 

600 

623 

647 

670 

694 

24 

185 

717 

741 

764 

788 

811 

834 

858 

881 

905 

928 

23 

186 

951 

975 

998 

o2l 

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1  38 

16! 

23 

187 

27184 

207 

231 

254 

277 

300 

323 

346 

370 

393 

23 

188 

416 

439 

462 

485 

508 

531 

554 

577 

600 

623 

23 

189 

646 

669 

692 

715 

738 

761 

784 

807 

830 

852 

23 

IT. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D, 

190-234 


LOGARITHMS. 


27875-3708^ 


N, 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

190 

27875 

898 

921 

944 

967 

989 

o!2 

o35 

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08! 

23 

191 

28103 

126 

149 

171 

194 

217 

240 

262 

285 

307 

23 

192 

330 

353 

375 

398 

421 

443 

466 

488 

511 

533 

23 

193 

556' 

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601 

623 

646 

668 

691 

713 

735 

758 

22 

194 

780 

803 

825 

847 

870 

892 

914 

937 

959 

981 

'22 

195 

29003 

026 

048 

070 

092 

115 

137 

159 

181 

203 

22 

196 

226 

248 

270 

292 

314 

336 

358 

380 

403 

425 

22 

197 

447 

469 

491 

513 

535 

5o< 

0/9 

601 

623 

645 

22 

198 

667 

688 

710 

732 

7.')  4 

776 

798 

820 

842 

863 

22 

199 

885 

907 

929 

951 

973 

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22 

200 

30103 

125 

146 

168 

190 

211 

233 

255 

276 

298 

22 

201 

320 

341 

363 

384 

406 

428 

449 

471 

492 

514 

22 

202 

535 

557 

578 

600 

621 

643 

664 

685 

707 

728 

21 

203 

750 

771 

792 

814 

835 

856 

878 

899 

920 

942 

21 

204 

963 

984 

o06 

o27 

o48 

o69 

o91 

i!2 

i33 

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21 

205 

31175 

197 

218 

239 

260 

281 

302 

323 

345 

366 

21 

206 

387 

408 

429 

450 

471 

492 

513 

534 

555 

576 

21 

207 

597 

618 

639 

660 

681 

702 

723 

744 

765 

785 

21 

208 

806 

827 

848 

869 

890 

911 

931 

952 

973 

994 

21 

209 

32015 

035 

056 

077 

098 

118 

139 

160 

181 

201 

21 

210 

222 

243 

263 

284 

305 

325 

346 

366 

387 

408 

21 

211 

428 

449 

469 

490 

510 

531 

552 

572 

593 

613 

20 

212 

634 

654 

675 

695 

715 

736 

756 

777 

797 

818 

20 

213 

838 

858 

879 

899 

919 

940 

960 

980 

oOl 

«.21 

20 

214 

33041 

062 

082 

102 

122 

143 

163 

183 

203 

224 

20 

215 

244 

264 

284 

304 

325 

345 

365 

385- 

405 

425 

20 

216 

445 

465 

486 

506 

526 

546 

566 

586 

606 

626 

20 

217 

646 

666 

686 

706 

726 

746 

766 

786 

806 

826 

20 

218 

846 

866 

885 

905 

925 

945, 

965 

985 

o05 

o25 

20 

219 

34044 

064 

084 

104 

124 

143 

163 

183 

203 

223 

20 

220 

242 

262 

282 

301 

321 

341 

361 

380 

400 

420 

20 

221 

439 

459 

479 

498 

518 

537 

557 

577 

596 

616 

20 

222 

635 

655 

674 

694 

713 

733 

753 

772 

792 

811 

19 

223 

830 

850 

869 

889 

908 

928 

947 

967 

986 

o05 

19 

224 

35025 

044 

064 

083 

102 

122 

141 

160 

180 

199 

19 

225 

218 

238 

257 

276 

295 

315 

334 

353 

372 

392 

19 

226 

.  411 

430 

449 

468 

488 

507 

526 

545 

564 

583 

19 

227 

603 

622 

641 

660 

679 

698 

717 

736 

755 

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19 

228 

793 

813 

832 

851 

870 

889 

908 

927 

946 

965 

19 

229 

984 

o03 

o21 

o40 

o59 

o78 

o97 

i!6 

i35 

i54 

19 

230 

36173 

192 

211 

229 

248 

267 

286 

305 

324 

342 

19 

231 

361 

380 

399 

418 

436 

455 

474 

493 

511 

530 

19 

232 

549 

568 

586 

605 

624 

642 

661 

680 

698 

717 

19 

233 

736 

754 

773 

791 

810 

829 

847 

866 

884 

903 

19 

234 

922 

940 

959 

977 

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o!4 

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o70 

088 

18 

N, 

0 

1  \ 

2 

3 

4 

5 

6 

7 

8 

9 

D, 

235-279 


LOGARITHMS. 


37107-44700 


I, 

0 

1 

2 

3 

4 

5 

6 

'  c 

8 

9 

D, 

235 

37107 

125 

144 

162 

181 

199 

218 

236 

254 

273 

18 

236 

291 

310 

328 

346 

365 

383 

401 

420 

438 

457 

18 

237 

475 

493 

511 

530 

548 

566 

585 

603 

621 

639 

18 

238 

658 

676 

694 

712 

731 

749 

767 

785 

803 

822 

18 

239 

840 

858 

876 

894 

912 

931 

949 

967 

985 

o03 

18 

240 

38021 

039 

057 

075 

093 

112 

130 

148 

166 

184 

18 

241 

202 

220 

238 

256 

274 

292 

310 

328 

346 

364 

18 

242 

382 

399 

417 

435 

453 

471 

489 

507 

525 

543 

18 

243 

561 

578 

596 

614 

632 

650 

668 

686 

703 

721 

18 

244 

739 

757 

775 

792 

810 

828 

846 

863 

881 

899 

18 

245 

917 

934 

952 

970 

987 

o05 

o23 

o41 

o58 

o76 

18 

246 

39094 

111 

129 

146 

164 

182 

199 

217 

235 

252 

18 

247 

270 

287 

305 

322 

340 

358 

375 

393 

410 

428 

18 

248 

445 

463 

480 

498 

515 

533 

550 

568 

585 

602 

18 

249 

620 

637 

655 

672 

690 

707 

724 

742 

759 

777 

17 

250 

794 

811 

829 

846 

863 

881 

898 

915 

933 

950 

17 

251 

967 

985 

o02 

o!9 

o37 

o54 

o71 

088 

i06 

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17 

252 

40140 

157 

175 

192 

209 

226 

243 

261 

278 

295 

17 

253 

312 

329 

346 

364 

381 

398 

415 

432 

449 

466 

17 

254 

483 

500 

518 

535 

552 

569 

586 

603 

620 

637 

17 

255 

654 

671 

688 

705 

722 

739 

756 

773 

790 

807 

17 

256 

824 

841 

858 

875 

892 

909 

926 

943 

960 

976 

17 

257 

993 

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o27 

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06! 

o78 

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i28 

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17 

258 

41162 

179 

196 

212 

229 

246 

263 

280 

296 

313 

17 

259 

--330 

347 

363 

380 

397 

414 

430 

447 

464 

481 

17 

260 

497 

514 

531 

547 

564 

581 

597 

614 

631 

647 

17 

261 

664 

681 

697 

714 

731 

747 

764 

780 

797 

814 

17 

262 

830 

847 

863 

880 

896 

913 

929 

946 

963 

979 

16 

263 

996 

o!2 

o29 

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o62 

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i27 

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16 

264 

42160 

177 

193 

210 

226 

243 

259 

275 

292 

308 

16 

265 

325 

341 

357 

374 

390 

406 

423 

439 

455 

472 

16 

266 

488 

504 

521 

537 

553 

570 

586 

602 

619 

635 

16 

267 

651 

667 

684 

700 

716 

732 

749 

765 

781 

797 

16 

268 

813 

830 

846 

862 

878 

894 

911 

927 

943 

959 

16 

269 

975 

991 

o08 

o24 

o40 

o56 

o72 

088 

i04 

i20 

16 

270 

43136 

152 

169 

185 

201 

217 

233 

249 

265 

281 

16 

271 

297 

313 

329 

345 

361 

377 

393 

409 

425 

441 

16 

272 

457 

473 

489 

505 

521 

537 

553 

569 

584 

600 

16 

273 

616 

632 

648 

664 

680 

696 

712 

727 

743 

759 

16 

274 

775 

791 

807 

823 

838 

854 

870 

886 

902 

917 

16 

275 

933 

949 

965 

981 

996 

o!2 

(.28 

o44 

o59 

o75 

16 

276 

44091 

107 

122 

138 

154 

170 

185 

201 

217 

232 

16 

277 

248 

264 

279 

295 

311 

326 

342 

358 

373 

389 

16 

278 

404 

420 

436 

451 

467 

483 

498 

514 

529 

545 

16 

279 

560 

576 

592 

607 

623 

638 

654 

669 

685 

700 

16 

If, 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D, 

S.  N.  35. 


280-324 


LOGARITHMS. 


47716-51175 


V, 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

280 

44716 

731 

747 

762 

778 

793 

809 

824 

840 

855 

15 

281 

871 

886 

902 

917 

932 

948 

963 

979 

994 

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15 

282 

45025 

040 

056  ^  071 

086 

102 

117 

133 

148 

163 

15 

283 

179 

194 

209 

225 

240 

255 

271 

286 

301 

317 

15 

284 

332 

347 

362 

378 

393 

408 

423 

439 

454 

469 

15 

285 

484 

500 

515 

530 

545 

561 

576 

591 

606 

621 

15 

286 

637 

652 

667 

682 

697 

712 

728 

743 

758 

773 

15 

287 

788 

803 

818. 

834 

849 

864 

879 

894 

909 

924 

15 

288 

939 

954 

969 

984 

oOO 

o!5 

o30 

o45 

o60 

o75 

15 

289 

46090 

105 

120 

135 

150 

165 

180 

195 

210 

225 

15 

290 

240 

255 

270 

285 

300 

315 

330 

345 

359 

374 

15 

291 

389 

404 

419 

434 

449 

464 

479 

494 

509 

523 

15 

292 

538 

553 

568 

583 

598 

613 

627 

642 

657 

672 

15 

293 

687 

702 

716 

731 

746 

761 

776 

790 

805 

820 

15 

294 

835 

850 

864 

879 

894 

909 

923 

938 

953 

967 

15 

295 

982 

997 

o!2 

o26 

o41 

o56 

o70 

o85 

lOO 

i!4 

15 

296 

47129 

144 

159 

173 

188 

202 

217 

232 

246 

261 

15 

297 

276 

290 

305 

319 

334 

349 

363 

378 

392 

407 

15 

298 

422 

436 

451 

465 

480 

494 

509 

524 

538 

553 

15 

299 

567 

582 

596 

611 

625 

640 

654 

669 

683 

698 

15 

300 

712 

727 

741 

756 

770 

784 

799 

813 

828 

842 

14 

301 

857 

871 

885 

900 

914 

929 

943 

958 

972 

986 

14 

302 

48001 

015 

029 

044 

058 

073 

087 

101 

116 

130 

14 

303 

144 

159 

173 

187 

202 

216 

230 

244 

259 

273 

14 

304 

287 

302 

316 

330 

344 

359 

373 

387 

401 

416 

14 

305 

430 

444 

458 

473 

487 

501 

515 

530 

544 

558 

14 

306 

572 

586 

601 

615 

629 

643 

657 

671 

686 

700 

14 

307 

714 

728 

742 

756 

770 

785 

799 

813 

827 

841 

14 

308 

855 

869 

883 

897 

911 

926 

940 

954 

968 

982 

14 

309 

996 

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066 

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14 

310 

49136 

150 

164 

178 

192 

206 

220 

234 

248 

262 

14 

311 

276 

290 

304 

318 

332 

346 

360 

374 

388 

402 

14 

312 

415 

429 

443 

457 

471 

485 

499 

513 

527 

541 

14 

313 

554 

568 

582 

596 

610 

624 

638 

651 

665 

679 

14 

314 

693 

707 

721 

734 

748 

762 

776 

790 

803 

817 

14 

315 

831 

845 

859 

872 

886 

900 

914 

927 

941 

955 

14 

316 

969 

982 

996 

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o37 

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14 

317 

50106 

120 

133 

147 

161 

174 

188 

202 

215 

229 

14 

318 

243 

256 

270 

284 

297 

311 

325 

338 

352 

365 

14 

319 

379 

393 

406 

420 

433 

447 

461 

474 

488 

501 

14 

320 

515 

529 

542 

556 

569 

583 

596 

610 

623 

637 

14 

321 

651 

664 

678 

691 

705 

718 

732 

745 

759 

772 

14 

322 

786 

799 

813 

826 

840 

853 

866 

880 

893 

907 

13 

323 

920 

934 

947 

961 

974 

987 

oOl 

o!4 

o28 

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13 

324 

51055 

068 

081 

095 

108 

121 

135 

148 

162 

175 

13 

N, 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

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325-369 


LOGARITHMS. 


51188-58808 


I, 

0 

1 

2 

3 

4 

5 

6 

7 

8 

g 

D. 

325 

51188 

202 

215 

228 

242 

255 

268 

282 

295 

308 

13 

326 

322 

335 

348 

362 

375 

388 

402 

415 

428 

441 

13 

327 

455 

468 

481 

495 

508 

521 

534 

548 

561 

574 

13 

328 

587 

601 

614 

627 

640 

654 

667 

680 

693 

706 

13 

329 

720 

733 

746 

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772 

786 

799 

812 

825 

838 

13 

330 

851 

865 

878 

891 

904 

917 

930 

943 

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970 

13 

331 

983 

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o09 

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lOl 

13 

332 

52114 

127 

140 

153 

166 

179 

192 

205 

218 

231 

13 

333 

244 

257 

270 

284 

297 

310 

323 

336 

349 

362 

13 

334 

375 

388 

401 

414 

427 

440 

453 

466 

479 

492 

13 

335 

504 

517 

530 

543 

556 

569 

582 

595 

608 

621 

13 

336 

634 

647 

660 

673 

686 

699 

711 

724 

737 

750 

13 

337 

763 

776 

789 

802 

815 

827 

840 

853 

866 

879 

13 

338 

892 

905 

917 

930 

943 

956 

969 

982 

994 

o07 

13 

339 

53020 

033 

046 

058 

071 

084 

097 

110 

122 

135 

13 

340 

148 

161 

173 

186 

199 

212 

224 

237 

250 

263 

13 

341 

275 

288 

301 

314 

326 

339 

352 

364 

377 

390 

13 

342 

403 

415 

428 

441 

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466 

479 

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517 

13 

343 

529 

542 

555 

567 

580 

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605 

618 

631 

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13 

344 

656 

668 

681 

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719 

732 

744 

757 

769 

13 

345 

782 

794 

807 

820 

832 

845 

857 

870 

882 

895 

13 

346 

908 

920 

933 

945 

958 

970 

983 

995 

o08 

o20 

13 

347 

54033 

045 

058 

070 

083 

095 

108 

120 

133 

145 

13 

348 

158 

170 

183 

195 

208 

220 

233 

245 

258 

270 

12 

349 

283 

295 

307 

320 

332 

345 

357 

370 

382 

394 

12 

350 

407 

419 

432 

444 

456 

469 

481 

494 

506 

518 

12 

351 

531 

543 

555 

568 

580 

593 

605 

617 

630 

642 

12 

352 

654 

667 

679 

691 

704 

716 

728 

741 

753 

765 

12 

353 

777 

790 

802 

814 

827 

839 

851 

864 

876 

888 

12 

354 

900 

913 

925 

937 

949 

962 

974 

986 

998 

oil 

12 

355 

55023 

035 

047 

060 

072 

084 

096 

108 

121 

133 

12 

356 

145 

157 

169 

182 

194 

206 

218 

230 

242 

255 

12 

357 

267 

279 

291 

303 

315 

328 

340 

352 

364 

376 

12 

358 

388 

400 

413 

425 

437 

449 

461 

473 

485 

497 

12 

359 

509 

522 

534 

546 

558 

570 

582 

594 

606 

618 

12 

360 

630 

642 

654 

666 

678 

691 

703 

715 

727 

739 

12 

361 

751 

763 

775 

787 

799 

811 

823 

835 

847 

859 

12 

362 

871 

883 

895 

907 

919 

931 

943 

955 

967 

979 

12 

363 

991 

o03 

o!5 

o27 

o38 

o50 

o62 

o74 

086 

o98 

12 

364 

56110 

122 

134 

146 

158 

170 

182 

194 

205 

217 

12 

365 

229 

241 

253 

265 

277 

289 

301 

312 

324 

336 

12 

366 

348 

360 

372 

384 

396 

407 

419 

431 

443 

455 

12 

367 

467 

478 

490 

502 

514 

526 

538 

549 

561 

573 

12 

368 

585 

597 

608 

620 

632 

644 

656 

667 

679 

691 

12 

369 

703 

714 

726 

738 

750 

761 

773 

785 

797 

808 

12 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D, 

370-414 


LOGARITHMS. 


56820-61794 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

37CT 

56820 

832 

844 

855 

867 

879 

891 

902 

914 

926 

12 

371 

937 

949 

961 

972 

984  - 

996 

o08 

o!9 

o31 

o43 

12 

372 

57054 

066 

078 

089 

101 

113 

124 

136 

148 

159 

12 

373 

171 

183 

194 

206 

217 

229 

241 

252 

264 

276 

12 

374 

287 

299 

310 

322 

334 

345 

357 

368 

380 

392 

12 

375 

403 

415 

426 

438 

449 

461 

473 

484 

496 

507 

12 

376 

519 

530 

542 

553 

565 

576 

588 

600 

611 

623 

12 

377 

634 

646 

657 

669 

680 

692 

703 

715 

726 

738 

11 

378 

749 

761 

772 

784 

795 

807 

818 

830 

841 

852 

11 

379 

864 

875 

887 

898 

910 

921 

933 

944 

955 

967 

11 

380 

978 

990 

oOl 

o!3 

o24 

o35 

o47 

o58 

o70 

08! 

11 

381 

58092 

104 

115 

127 

138 

149 

161 

172 

184 

195 

11 

382 

206 

218 

229 

240 

252 

263 

274 

286 

297 

309 

11 

383 

320 

331 

343 

354 

365 

377 

388 

399 

410 

422 

11 

384 

433 

444 

456 

467 

478 

490 

501 

512 

524 

535 

11 

385 

546 

557 

569 

580 

591 

602 

614 

625 

636 

647 

11 

386 

659 

670 

681 

692 

704 

715 

726 

737 

749 

760 

11 

387 

771 

782 

794 

805 

816 

827 

838 

850 

861 

872 

11 

388 

883 

894 

906 

917 

928 

939 

950 

961 

973 

984 

11 

389 

995 

o06 

o!7 

o28 

040 

o51 

o62 

o73 

o84 

o95 

11 

390 

59106 

118 

129 

140 

151 

162 

173 

184 

195 

207 

11 

391 

218 

229 

240 

251 

262 

273 

284 

295 

306 

318 

11 

392 

329 

340 

351 

362 

373 

384 

395 

406 

417 

428 

11 

393 

439 

450 

461 

472 

483 

494 

506 

517 

528 

539 

11 

394 

550 

561 

572 

583 

594 

605 

616 

627 

638 

649 

11 

395 

660 

671 

682 

693 

704 

715 

726 

737 

748 

759 

11 

396 

770 

780 

791 

802 

813 

824 

835 

846 

857 

868 

11 

397 

879 

890 

901 

912 

923 

934 

945 

956 

966 

977 

11 

398 

988 

999 

olO 

o21 

o32 

o43 

o54 

060 

o76 

086 

11 

399 

60097 

108 

119 

130 

141 

152 

-163 

173 

184 

195 

11 

400 

206 

217 

228 

239 

249 

260 

271 

282 

293  i  304 

11 

401 

314 

325 

336 

347 

358 

369 

379 

390 

401 

412 

11 

402 

423 

433 

444 

455 

466 

477 

487 

498 

509 

520 

11 

403 

531 

541 

552 

563 

574 

584 

595 

606 

617 

627 

11 

404 

638 

649 

660 

670 

681 

692 

703 

713 

724 

735 

11 

405 

746 

756 

767 

778 

788 

799 

810 

821 

831 

842 

11 

406 

853 

863 

874 

885 

895 

906 

917 

927 

938 

949 

11 

407 

959 

970 

981 

991 

o02 

o!3 

o23 

o34 

o45 

o55 

11 

408 

61066 

077 

087 

098 

109 

119 

130 

140 

151 

162 

11 

409 

172 

183 

194 

204 

215 

225 

236 

247 

257 

268 

11 

410 

278 

289 

300 

310 

321 

331 

342 

352 

363 

374 

11 

411 

384 

395 

405 

416 

426 

437 

448 

458 

469 

479 

11 

412 

490 

500 

511 

521 

532 

542 

553 

563 

574 

584 

11 

413 

595 

606 

616 

627 

637 

648 

658 

669 

679 

690 

11 

414 

700 

711 

721 

731 

742 

752 

763 

773 

784 

794 

10 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

415-459 


LOGARITHMS. 


61805-662G6 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D, 

415 

61805 

815 

826 

836 

847 

857 

868 

878 

888 

899 

10 

416 

909 

920 

930 

941 

951 

962 

972 

982 

993 

o03 

10 

417 

62014 

024 

034 

045 

055 

066 

076 

086 

097 

107 

10 

418 

118 

128 

138 

149 

159 

170 

180 

190 

201 

211 

10 

419 

221 

232 

242 

252 

263 

273 

284 

294 

304 

315 

10 

420 

325 

335 

346 

356 

366 

377 

387 

397 

408 

418 

10 

421 

428 

435 

449 

459 

469 

480 

490 

500 

511 

521 

10 

422 

531 

542 

552 

562 

572 

583 

593 

603 

613 

624 

10 

423 

634 

644 

655 

665 

675 

685 

696 

706 

716 

726 

10 

424 

737 

747 

757 

767 

778 

788 

798 

808 

818 

829 

10 

425 

839 

849 

859 

870 

880 

890 

900 

910 

921 

931 

10 

426 

941 

951 

961 

972 

982 

992 

o02 

o!2 

o22 

o33 

10 

427 

63043 

053 

063 

073 

083 

094 

104 

114 

124 

134 

10 

428 

144 

155 

165 

175 

185 

195 

205 

215 

225 

236 

10 

429 

246 

256 

266 

276 

286 

296 

306 

317 

327 

337 

10 

430 

347 

357 

367 

377 

387 

397 

407 

417 

428 

438 

10 

431 

448 

458 

468 

478 

488 

498 

508 

518 

528 

538 

10 

432 

548 

558 

568 

579 

589 

599 

609 

619 

629 

639 

10 

433 

649 

659 

669 

679 

689 

699 

709 

719 

729 

739 

10 

434 

749 

759 

769 

779 

789 

799 

809 

819 

829 

839 

10 

435 

849 

859 

869 

879 

889 

899 

909 

919 

929 

939 

10 

436 

949 

959 

969 

979 

988 

998 

o08 

o!8 

o28 

o38 

10 

437 

64048 

058 

068 

078 

088 

098 

108 

118 

128 

137 

10 

438 

147 

157 

167 

177 

187 

197 

207 

217 

227 

237 

10 

439 

246 

256 

266 

276 

286 

296 

306 

316 

326 

335 

10 

440 

345 

355 

365 

375 

385 

395 

404 

414 

424 

434 

10 

441 

444 

454 

464 

473 

483 

493 

503 

513 

523 

532 

10 

442 

542 

552 

562 

572 

582 

591 

601 

611 

621 

631 

10 

443 

640 

650 

660 

670 

680 

689 

699 

709 

719 

729 

10 

444 

738 

748 

758 

768 

777 

787 

797 

807 

816 

826 

10 

445 

836 

846 

856 

865 

875 

885 

895 

904 

914 

924 

10 

446 

933 

943 

953 

963 

972 

982 

992 

o02 

oil 

o21 

10 

447 

65031 

040 

050 

060 

070 

079 

089 

099 

108 

118 

10 

448 

128 

137 

147 

157 

167 

176 

186 

196 

205 

215 

10 

449 

225 

234 

244 

254 

263 

273 

283 

292 

302 

312 

10 

450 

321 

331 

341 

350 

360 

369 

379 

389 

398 

408 

10 

451 

418 

427 

437 

447 

456 

466 

475 

485 

495 

504 

10 

452 

514 

523 

533 

543 

552 

562 

571 

581 

591 

600 

10 

453 

610 

619 

629 

639 

648 

658 

667 

677 

686 

696 

10 

454 

706 

715 

725 

734 

744 

753 

763 

772 

782 

792 

9 

455 

801 

811 

820 

830 

839 

849 

858 

868 

877 

887 

9 

456 

896 

906 

916 

925 

935 

944 

954 

963 

973 

982 

9 

457 

992 

oOl 

oil 

o20 

o30 

o39 

o49 

o58 

068 

o77 

9 

458 

66087 

096 

106 

115 

124 

134 

143 

153 

162 

172 

9 

459 

181 

191 

200 

210 

219 

229 

238 

247 

257 

266 

9 

N, 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D, 

430-504 


LOGARITHMS. 


66276-70321 


N, 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

460 

66276 

285 

295 

304 

314 

323 

332 

342 

351 

361 

9 

461 

370 

380 

389 

398 

408 

417 

427 

436 

445 

455 

9 

462 

464 

474 

483 

492 

502 

511 

521 

530 

539 

549 

9 

463 

558 

567 

577 

586 

596 

605 

614 

624 

633 

642 

9 

464 

652 

661 

671 

680 

689 

699 

708 

717 

727 

736 

9 

465 

745 

755 

764 

773 

783 

792 

801 

811 

820 

829 

9 

466 

839 

848 

857 

867 

876 

885 

894 

904 

913 

922 

9 

467 

932 

941 

950 

960 

969 

978 

987 

997 

o06 

o!5 

9 

468 

67025 

034 

043 

052 

062 

071 

080 

089 

099 

108 

9 

469 

117 

127 

136 

145 

154 

164 

173 

182 

191 

201 

9 

470 

210 

219 

228 

237 

247 

256 

265 

274 

284 

293 

9 

471 

302 

311 

321 

330 

339 

348 

357 

367 

376 

385 

9 

472 

394 

403 

413 

422 

431 

440 

449 

459 

468 

477 

9 

473 

486 

495 

504 

514 

523 

532 

541 

550 

560 

569 

9 

474 

578 

587 

596 

605 

614 

624 

633 

642 

651 

660 

9 

475 

669 

679 

688 

697 

706 

715 

724 

733 

742 

752 

9 

476 

761 

770 

779 

788 

797 

806 

815 

825 

834 

843 

9 

477 

852 

861 

870 

879 

888 

897 

906 

916 

925 

934 

9 

478 

943 

952 

961 

970 

979 

988 

997 

o06 

o!5 

o24 

9 

479 

68034 

043 

052 

061 

070 

079 

088 

097 

106 

115 

9 

480 

124 

133 

142 

151 

160 

169 

178 

187 

196 

205 

9 

481 

215 

224 

233 

242 

251 

260 

269 

278 

287 

296 

9 

482 

305 

314 

323 

332 

341 

350 

359 

368 

377 

386 

9 

483 

395 

404 

413 

422 

431 

440 

449 

458 

467 

476 

9 

484 

485 

494 

502 

511 

520 

529 

538 

547 

556 

565 

9 

485 

574 

583 

592 

601 

610 

619 

628 

637 

646 

655 

9 

486 

664 

673 

681 

690 

699 

708 

717 

726 

735 

744 

9 

487 

753 

762 

771 

780 

789 

797 

806 

815 

824 

833 

9 

488 

842 

851 

860 

869 

878 

886 

895 

904 

913 

922 

9 

489 

931 

940 

949 

958 

966 

975 

984 

993 

o02 

oil 

9 

490 

69020 

028 

037 

046 

055 

064 

073 

082 

090 

099 

9 

491 

108 

117 

126 

135 

144 

152 

161 

170 

179 

188 

9 

492 

197 

205 

214 

223 

232 

241 

249 

258 

267 

276 

9 

493 

285 

294 

302 

311 

320 

329 

338 

346 

355 

364 

9 

494 

373 

381 

390 

399 

408 

417 

425 

434 

443 

452 

9 

495 

461 

469 

478 

487 

496 

504 

513 

522 

531 

539 

9 

496 

548 

557 

566 

574 

583 

592 

601 

609 

618 

627 

9 

497 

636 

644 

653 

662 

671 

679 

688 

697 

705 

714 

9 

498 

723 

732 

740 

749 

758 

767 

775 

784 

793 

801 

9 

499 

810 

819 

827 

836 

845 

854 

862 

871 

880 

888 

9 

500 

897 

906 

914 

923 

932 

940 

949 

958 

966 

975 

9 

501 

984 

992 

oOl 

olO 

o!8 

o27 

o36 

o44 

o53 

o62 

9 

502 

70070 

079 

088 

096 

105 

114 

122 

131 

140 

148 

9 

503 

157 

165 

174 

183 

191 

200 

209 

217 

226 

234 

9 

504 

243 

252 

260 

269 

278 

286 

295 

303 

312 

321 

9 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D, 

505-549 


LOGARITHMS. 


70329-74028 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D, 

505 

70329 

338 

346 

355 

364 

372 

381 

389 

398 

406 

9 

506 

415 

424 

432 

441 

449 

458 

467 

475 

484 

492 

9 

507 

501 

509 

518 

526 

535 

544 

552 

561 

569 

578 

9 

508 

586 

595 

603 

612 

621 

629 

638 

646 

655 

663 

9 

509 

672 

680 

689 

697 

706 

714 

723 

731 

740 

749 

9 

510 

757 

766 

774 

783 

791 

800 

808 

817 

825 

834 

9 

511 

842 

851 

859 

868 

876 

885 

893 

902 

910 

919 

9 

512 

927 

935 

944 

952 

961 

969 

978 

986 

995 

o03 

9 

513 

71012 

020 

029 

037 

046 

054 

063 

071 

079 

088 

8 

514 

096 

105 

113 

122 

130 

139 

147 

155 

164 

172 

8 

515 

181 

189 

198 

206 

214 

223 

231 

240 

248 

257 

8 

516 

265 

273 

282 

290 

299 

307 

315 

324 

332 

341 

8 

517 

349 

357 

•366 

374 

383 

391 

399 

408 

416 

425 

8 

518 

433 

441 

450 

458 

466 

475 

483 

492 

500 

508 

8 

519 

517 

525 

533 

542 

550 

559 

567 

575 

584 

592 

8 

520 

600 

609 

G17 

625. 

634 

642 

650 

659 

667 

675 

8 

521 

684 

692 

700 

709 

717 

725 

734 

742 

750 

759 

8 

522 

767 

775 

784 

792 

800 

809 

817 

825 

834 

842 

8 

523 

850 

858 

867 

875 

883 

892 

900 

908 

917 

925 

8 

524 

933 

941 

950 

958 

966 

975 

983 

991 

999 

o08 

8 

525 

72016 

024 

032 

041 

049 

057 

066 

074 

082 

090 

8 

526 

099 

107 

115 

123 

132 

140 

148 

156 

165 

173 

8 

527 

181 

189 

198 

206 

214 

222 

230 

239 

247 

255 

8 

528 

263 

272 

280 

288 

296 

304 

313 

321 

329 

337 

8 

529 

346 

354 

362 

370 

378 

387 

395 

403 

411 

419 

8 

530 

428 

436 

444 

452 

460 

469 

477 

485 

493 

501 

8 

531 

509 

518 

526 

534 

542 

550 

558 

567 

575 

583 

8 

532 

591 

599 

607 

61-6 

624 

632 

640 

648 

656 

665 

8 

533 

673 

681 

689 

697 

705 

713 

722 

730 

738 

746 

8 

534 

754 

762 

770 

779 

787 

795 

803 

811 

819 

827 

8 

535 

835 

843 

852 

860 

868 

876 

884 

892 

900 

908 

8 

536 

916 

925 

933 

941 

949 

957 

965 

973 

981 

989 

8 

537 

997 

o06 

o!4 

o22 

o30 

n  3  8 

o46 

o54 

o62 

o70 

8 

538 

73078 

086 

094 

102 

111 

119 

127 

135 

143 

151 

8 

539 

159 

167 

175 

183 

191 

199 

207 

215 

223 

231 

8 

540 

239 

247 

255 

263 

272 

280 

288 

296 

304 

312 

8 

541 

320 

328 

336 

344 

352 

360 

368 

376 

384 

392 

8 

542 

400 

408 

416 

424 

432 

440 

448 

456 

464 

472 

8 

543 

480 

488 

496 

504 

512 

520 

528 

536 

544 

552 

8 

544 

560 

568 

576 

584 

592 

600 

608 

616 

624 

632 

8 

545 

640 

648 

656 

664 

672 

679 

687 

695 

703 

711 

8 

546 

719 

727 

735 

743 

751 

759 

767 

775 

783 

791 

8 

547 

799 

807 

815 

823 

830 

838 

846 

854 

862 

870 

8 

548 

878 

886 

894 

902 

910 

918 

926 

933 

941 

949 

8 

549 

957 

965 

973 

981 

989 

997 

o05 

o!3 

o2() 

o28 

8 

N, 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

11 


550-594 


LOGARITHMS. 


74036-77444 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

550 

74036 

044 

052 

060 

068 

076 

084 

092 

099 

107 

8 

551 

115 

123 

131 

139 

147 

155 

162 

170 

178 

186 

8 

552 

194 

202 

210 

218 

225 

233 

241 

249 

257 

265 

8 

553 

273 

280 

288 

296 

304 

312 

320 

327 

335 

343 

8 

554 

351 

359 

367 

374 

382 

390 

398 

406 

414 

421 

8 

555 

429 

437 

445 

453 

461 

468 

476 

484 

492 

500 

8 

556 

507 

515 

523 

531 

539 

547 

554 

562 

570 

578 

8 

557 

586 

593 

601 

609 

617 

624 

632 

640 

648 

656 

8 

558 

663 

671 

679 

687 

695 

702 

710 

718 

726 

733 

8 

559 

741 

749 

757 

764 

772 

780 

788 

796 

803 

811 

8 

560 

819 

827 

834 

842 

850 

858 

865 

873 

881 

889 

8 

561 

896 

904 

912 

920 

927 

935 

943 

950 

958 

966 

8 

562 

974 

981 

989 

997 

o05 

o!2 

o20 

o28 

o35 

o43 

8 

563 

75051 

059 

066 

074 

082 

089 

097 

105 

113 

120 

8 

564 

128 

136 

143 

151 

159 

166 

174 

182 

.189 

197 

8 

565 

205 

213 

220 

228 

236 

243 

251 

259 

266 

274 

8 

566 

282 

289 

297 

305 

312 

320 

328 

335 

343 

351 

8 

567 

358 

366 

374 

381 

389 

397 

404 

412 

420 

427 

8 

568 

435 

442 

450 

458 

465 

473 

481 

488 

496 

504 

8 

'569 

511 

519 

526 

534 

542 

549 

557 

565 

572 

580 

8 

570 

587 

595 

603 

610 

618 

626 

633 

641 

648 

656 

8 

571 

664 

671 

679 

686 

694 

702 

709 

717 

724 

732 

8 

572 

740 

747 

755 

762 

770 

778 

785 

793 

800 

808 

8 

573 

815 

823 

831 

838 

846 

853 

861 

868 

876 

884 

8 

574 

891 

899 

906 

914 

921 

929 

937 

944 

952 

959 

8 

575 

967 

974 

982 

989 

997 

o05 

o!2 

o20 

o27 

o35 

8 

576 

76042 

050 

057 

065 

072 

080 

087 

095 

103 

110 

8 

577 

118 

125 

133 

140 

148 

155 

163 

170 

178 

185 

8 

578 

193 

200 

208 

215 

223 

230 

238 

245 

253 

260 

8 

579 

268 

275 

283 

290 

298 

305 

313 

320 

328 

335 

8 

580 

343 

350 

358 

365 

373 

380 

388 

395 

403 

410 

8 

581 

418 

425 

433 

440 

448 

455 

462 

470 

477 

485 

7 

582 

492 

500 

507 

515 

522 

530 

537 

545 

552 

559 

7 

583 

567 

574 

582 

589 

597 

604 

612 

619 

626 

634 

7 

584 

641 

649 

656 

664 

671 

678 

686 

693 

701 

708 

7 

585 

716 

723 

730 

738 

745 

753 

760 

768 

775 

782 

7 

586 

790 

797 

805 

812 

819 

827 

834 

842 

849 

856 

7 

587 

864 

871 

879 

886 

893 

901 

908 

916 

923 

930 

7 

588 

938 

945 

953 

960 

967 

975 

982 

989 

997 

o04 

7 

589 

77012 

019 

026 

034 

041 

048 

056 

063 

070 

078 

7 

590 

085 

093 

100 

107 

115 

122 

129 

137 

144 

151 

7 

591 

159 

166 

173 

181 

188 

195 

203 

210 

217 

225 

7 

592 

232 

240 

247 

254 

262 

269 

276 

283 

291 

298 

7 

593 

305 

313 

320 

327 

335 

342 

349 

357 

364 

371 

7 

594 

379 

386 

393 

401 

408 

415 

422 

430 

437 

444 

7 

N, 

0 

1 

2 

3 

4 

5 

6 

7 

8 

g 

D, 

12 


595-639 


LOGARITHMS. 


77452-80611 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D, 

595 

77452 

459 

466 

474 

481 

488 

495 

503 

510 

517 

7 

596 

525 

532 

539 

546 

554 

561 

568 

576 

583 

590 

7 

597 

597 

605 

612 

619 

627 

634 

641 

648 

656 

663 

7 

598 

670 

677 

685 

692 

699 

706 

714 

721 

728 

735 

7 

599 

743 

750 

757 

764 

772 

779 

786 

793 

801 

808 

7 

600 

815 

822 

830 

837 

844 

851 

859 

866 

873 

880 

7 

601 

887 

895 

902 

909 

916 

924 

931 

938 

945 

952 

7 

602 

960 

967 

974 

981 

988 

996 

o03 

olO 

o!7 

o25 

7 

603 

78032 

039 

046 

053 

061 

068 

075 

082 

089 

097 

7 

604 

104 

111 

118 

125 

132 

140 

147 

154 

161 

168 

7 

605 

176 

183 

190 

197 

204 

211 

219 

226 

233 

240 

7 

606 

247 

254 

262 

269 

276 

283 

290 

297 

305 

312 

7 

607 

319 

326 

333 

340 

347 

355 

362 

369 

376 

383 

7 

608 

390 

398 

405 

412 

419 

426 

433 

440 

447 

455 

7 

609 

462 

469 

476 

483 

490 

497 

504 

512 

519 

526 

7 

610 

533 

540 

547 

554 

561 

569 

576 

583 

590 

597 

7 

611 

604 

611 

618 

625 

633 

640 

647 

654 

661 

668 

7 

612 

675 

682 

689 

696 

704 

711 

718 

725 

732 

739 

7 

613 

746 

753 

760 

767 

774 

781 

789 

796 

803 

810 

7 

614 

817 

824 

831 

838 

845 

852 

859 

866 

873 

880 

7 

615 

888 

895 

902 

909 

916 

923 

930 

937 

944 

951 

7 

6J6 

958 

965 

972 

979 

986 

993 

oOO 

o07 

o!4 

o21 

7 

617 

79029 

036 

043 

050 

057 

064 

071 

078 

085 

092 

7 

618 

099 

106 

113 

120 

127 

134 

141 

148 

155 

162 

7 

619 

169 

176 

183 

190 

197 

204 

211 

218 

225 

232 

7 

620 

239 

246 

253 

260 

267 

274 

281 

288 

295 

302 

7 

621 

309 

316 

323 

330 

337 

344 

351 

358 

365 

372 

7 

622 

379 

386 

393 

400 

407 

414 

421 

428 

435 

442 

7 

623 

449 

456 

463 

470 

477 

484 

491 

498 

505 

511 

7 

624 

518 

525 

532 

539 

546 

553 

560 

567 

574 

581 

7 

625 

588 

595 

602 

609 

616 

623 

630 

637 

644 

650 

7 

626 

657 

664 

671 

678 

685 

692 

699 

706 

713 

720 

7 

627 

727 

734 

741 

748 

754 

761 

768 

775 

782 

789 

7 

628 

796 

803 

810 

817 

824 

831 

837 

844 

851 

858 

7 

629 

865 

872 

879 

886 

893 

900 

906 

913 

920 

927 

7 

630 

934 

941 

948 

955 

962 

969 

975 

982 

989 

996 

7 

631 

80003 

010 

017 

024 

030 

037 

044 

051 

058 

065 

7 

632 

072 

079 

085 

092 

099 

106 

113 

120 

127 

134 

7 

633 

140 

147 

154 

161 

168 

175 

182 

188 

195 

202 

7 

634 

209 

216 

223 

229 

236 

243 

250 

257 

264 

271 

7 

635 

277 

284 

291 

298 

305 

312 

318 

325 

332 

339 

7 

636 

346 

353 

359 

366 

373 

380 

387 

393 

400 

407 

7 

637 

414 

421 

428 

434 

441 

448 

455 

462 

468 

475 

7 

638 

482 

489 

496 

502 

509 

516 

523 

530 

536 

543 

7 

639 

550 

557 

564 

570 

577 

584 

591 

598 

604 

611 

7 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D, 

13 


649-684 


LOGARITHMS. 


80618-83563 


t. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

g 

D. 

640 

80618 

625 

632 

638 

645 

652 

659 

665 

672 

679 

7 

641 

686 

693 

699 

706 

713 

720 

726 

733 

740 

747 

7 

642 

754 

760 

767 

774 

781 

787 

794 

801 

808 

814 

7 

643 

821 

828 

835 

841 

848 

855 

862 

868 

875 

882 

7 

644 

889 

895 

902 

909 

916 

922 

929 

936 

943 

949 

7 

645 

956 

963 

969 

976 

983 

990 

996 

o03 

olO 

o!7 

7 

646 

81023 

030 

037 

043 

050 

057 

064 

070 

077 

084 

7 

647 

090 

097 

104 

111 

117 

124 

131 

137 

144 

151 

7 

648 

158 

164 

171 

178 

184 

191 

198 

204 

211 

218 

7 

649 

224 

231 

238 

245 

251 

258 

265 

271 

278 

285 

7 

650 

291 

298 

305 

311 

318 

325 

331 

338 

345 

351 

7 

651 

358 

365 

371 

378 

385 

391 

398 

405 

411 

418 

7 

652 

425 

431 

438 

445 

451 

458 

465 

471 

478 

485 

7 

653 

491 

498 

505 

511 

518 

525 

531 

538 

544 

551 

7 

654 

558 

564 

571 

578 

584 

591 

598 

604 

611 

617 

7 

655 

624 

631 

637 

644 

651 

657 

664 

671 

677 

684 

7 

656 

690 

697 

704 

710 

717 

723 

730 

737 

743 

750 

7 

657 

757 

763 

770 

776 

783 

790 

796 

803 

809 

816 

7 

658 

823 

829 

836 

842 

849 

856 

862 

869 

875 

882 

7 

659 

889 

895 

902 

908 

915 

921 

928 

935 

941 

948 

7 

660 

954 

961 

968 

974 

981 

987 

994 

oOO 

o07 

o!4 

7 

661 

82020 

027 

033 

040 

046 

053 

060 

066 

073 

079 

7 

662 

086 

092 

099 

105 

112 

119 

125 

132 

138 

145 

7 

663 

151 

158 

164 

171 

178 

184 

191 

197 

204 

210 

7 

664 

217 

223 

230 

236 

243 

249 

256 

263 

269 

276 

7 

665 

282 

289 

295 

302 

308 

315 

321 

328 

334 

'341 

7 

666 

347 

354 

360 

367 

373 

380 

387 

393 

400 

406 

7 

667 

413 

419 

426 

432 

439 

445 

452 

458 

465 

471 

7 

668 

478 

484 

491 

497 

504 

510 

517 

523 

530 

536 

7 

669 

543 

549 

556 

562 

569 

575 

582 

588 

595 

601 

7 

670 

607 

614 

620 

627 

633 

640 

646 

653 

659 

666 

7 

671 

672 

679 

685 

692 

698 

705 

711 

718 

724 

730 

6 

672 

737 

743 

750 

756 

763 

769 

776 

782 

789 

795 

6 

673 

802 

808 

814 

821 

827 

834 

840 

847 

853 

860 

6. 

674 

866 

872 

879 

885 

892 

898 

905 

911 

918 

924 

6 

675 

930 

937 

943 

950 

956 

963 

969 

975 

982 

988 

6 

676 

995 

oOl 

o08 

o!4 

o20 

o27 

o33 

o40 

o46 

o52 

6 

677 

83059 

065 

072 

078 

085 

091 

097 

104 

110 

117 

6 

678 

123 

129 

136 

142 

149 

155 

161 

168 

174 

181 

6 

679 

187 

193 

200 

206 

213 

219 

225 

232 

238 

245 

6 

680 

251 

257 

264 

270 

276 

283 

289 

296 

302 

308 

6 

681 

315 

321 

327 

334 

340 

347 

353 

359 

366 

372 

6 

682 

378 

385 

391 

398 

404 

410 

417 

423 

429 

436 

6 

683 

442 

448 

455 

461 

467 

474 

480 

487 

493 

499 

6 

684 

506 

512 

518 

525 

531 

537 

544 

550 

556 

563 

6 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D, 

14 


685-729 


LOGARITHMS. 


83569-86323 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

g 

D. 

685 

83569 

575 

582 

588 

594 

601 

607 

613 

620 

626 

6 

686 

632 

639 

645 

651 

658 

664 

670 

677 

683 

689 

6 

687 

696 

702 

70S 

715 

721 

727 

734 

740 

746 

753 

6 

688 

759 

765 

771 

778 

784 

790 

797 

803 

809 

816 

6 

689 

822 

828 

835 

841 

847 

853 

860 

866 

872 

879 

6 

690 

885 

891 

897 

904 

910 

916 

923 

929 

935 

942 

6 

691 

948 

954 

960 

967 

973 

979 

985 

992 

998 

o04 

6 

692 

84011 

017 

023 

029 

036 

042 

048 

055 

061 

067 

6 

693 

073 

080 

-086 

092 

098 

105 

111 

117 

123 

130 

6 

694 

136 

142 

148 

155 

161 

167 

173 

180 

186 

192 

6 

695 

198 

205 

211 

217 

223 

230 

236 

242 

248 

255 

6 

696 

261 

267 

273 

280 

286 

292 

298 

305 

311 

317 

6 

697 

323 

330 

336 

342 

348 

354 

361 

367 

373 

379 

6 

698 

386 

392 

398 

404 

410 

417 

423 

429 

435 

442 

6 

699 

448 

454 

460 

466 

473 

479 

485 

491 

497 

504 

6 

700 

510 

516 

522 

528 

535 

541 

547 

553 

559 

566 

6 

701 

572 

578 

584 

590 

597 

603 

609 

615 

621 

628 

6 

702 

634 

640 

646 

652 

658 

665 

671 

677 

683 

689 

6 

703 

696 

702 

708 

714 

720 

726 

733 

739 

745 

751 

6 

704 

757 

763 

770 

776 

782 

788 

794 

800 

807 

813 

6 

705 

819 

825 

831 

837 

844 

850 

856 

862 

868 

874 

6 

706 

880 

887 

893 

899 

905 

911 

917 

924 

930 

936 

6 

707 

942 

948 

9"54 

960 

967 

973 

979 

985 

991 

997 

6 

708 

85003 

009 

016 

022 

028 

034 

040 

046 

052 

058 

6 

709 

065 

071 

077 

083 

089 

095 

101 

107 

114 

120 

6 

710 

126 

132 

138 

144 

150 

156 

163 

169 

175 

181 

6 

711 

187 

193 

199 

205 

211 

217 

224 

230 

236 

242 

6 

712 

248 

254 

260 

266 

272 

278 

285 

291 

297 

303 

6 

713 

309 

315 

321 

327 

333 

339 

345 

352 

358 

364 

6 

714 

370 

376 

382 

388 

394 

400 

406 

412 

418 

425 

6 

715 

431 

437 

443 

449 

455 

461 

467 

473 

479 

485 

6 

716 

491 

497 

503 

509 

516 

522 

528 

534 

540 

546 

6 

717 

552 

558 

564 

570 

576 

582 

588 

594 

600 

606 

6 

718 

612 

618 

625 

631 

637 

643 

649 

655 

661 

667 

6 

719 

673 

679 

685 

691 

697 

703 

709 

715 

721 

727 

6 

720 

733 

739 

745 

751 

757 

763 

769 

775 

781 

788 

6 

721 

794 

800 

806 

812 

818 

824 

830 

836 

842 

848 

6 

722 

854 

860 

866 

872 

878 

884 

890 

896 

902 

908 

6 

723 

914 

920 

926 

932 

938 

944 

950 

956 

962 

968 

6 

724 

974 

980 

986 

992 

998 

o04 

oJO 

o!6 

o22 

o28 

6 

725 

86034 

040 

046 

052 

058 

064 

070 

076 

082 

088 

6 

726 

094 

100 

106 

112 

118 

124 

130 

136 

141 

147 

6 

727 

153 

159 

165 

171 

177 

183 

189 

195 

201 

207 

6 

728 

213 

219 

225 

231 

237 

243 

249 

255 

261 

267 

6 

729 

273 

279 

285 

291 

297 

303 

308 

314 

320 

326 

6 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D, 

15 


730-774 


LOGARITHMS. 


86332-88925 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

730 

86332 

338 

344 

350 

356 

362 

368 

374 

380 

386 

6 

731 

392 

398 

404 

410 

415 

421 

427 

433 

439 

445 

6 

732 

451 

457 

463 

469 

475 

481 

487 

493 

499 

504 

6 

733 

510 

516 

522 

528 

534 

540 

546 

552 

558 

564 

6 

734 

570 

576 

581 

587 

593 

599 

605 

611 

617 

623 

6 

735 

629 

635 

641 

646 

652 

658 

664 

670 

676 

682 

6 

736 

688 

694 

700 

705 

711 

717 

723 

729 

735 

741 

6 

737 

747 

753 

759 

764 

770 

776 

782 

788 

794 

800 

6 

738 

806 

812 

817 

823 

829 

835 

841 

847 

853 

859 

6 

739 

864 

870 

876 

882 

888 

894 

900 

906 

911 

917 

6 

740 

923 

929 

935 

941 

947 

953 

958 

964 

970 

976 

6 

741 

982 

988 

994 

999 

o05 

oil 

o!7 

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o29 

o35 

6 

742 

87040 

046 

052 

058 

064 

070 

075 

081 

087 

093 

6 

743 

099 

105 

111 

116 

122 

128 

134 

140 

146 

151 

6 

744 

157 

163 

169 

175 

181 

186 

192 

198 

204 

210 

6 

745 

216 

221 

227 

233 

239 

245 

251 

256 

262 

268 

6 

746 

274 

280 

286 

291 

297 

303 

309 

315 

320 

326 

6 

747 

332 

338 

344 

349 

355 

361 

367 

373 

379 

384 

6 

748 

390 

396 

402 

408 

413 

419 

425 

431 

437 

442 

6 

749 

448 

454 

460 

466 

471 

477 

483 

489 

495 

500 

6 

750 

506 

512 

518 

523 

529 

535 

541 

547 

552 

558 

6 

751 

564 

570 

576 

581 

587 

593 

599 

604 

610 

616 

G 

752 

622 

628 

633 

639 

645 

651 

656 

"662 

668 

674 

6 

753 

679 

685 

691 

697 

703 

708 

714 

720 

726 

731 

6 

754 

737 

743 

749 

754 

760 

766 

772 

777 

783 

789 

6 

755 

795 

800 

806 

812 

818 

823 

829 

835 

841 

846 

6 

756 

852 

858 

864 

869 

875 

881 

887 

892 

898 

904 

6 

757 

910 

915 

921 

927 

933 

938 

944 

950 

955 

961 

6 

758 

967 

973 

978 

984 

990 

996 

oOl 

o07 

oJ3 

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6 

759 

88024 

030 

036 

041 

047 

053 

058 

064 

070 

076 

6 

760 

081 

087 

093 

098 

104 

110 

116 

121 

127 

133 

6 

761 

138 

144 

150 

156 

161 

167 

173 

178 

184 

190 

6 

762 

195 

201 

207 

213 

218 

224 

230 

235 

241 

247 

6 

763 

252 

258 

264 

270 

275 

281 

287 

292 

298 

304 

6 

764 

309 

3J5 

321 

326 

332 

338 

343 

349 

355 

360 

G 

765 

366 

372 

377 

383 

389 

395 

400 

406 

412 

417 

6 

766 

423 

429 

434 

440 

446 

451 

457 

463 

468 

474 

6 

767 

480 

485 

491 

497 

502 

508 

513 

519 

525 

530 

6 

768 

536 

542 

547 

553 

559 

564 

570 

576 

581 

587 

6 

769 

593 

598 

604 

610 

615 

621 

627 

632 

638 

643 

G 

770 

649 

655 

660 

666 

672 

677 

683 

689 

694 

700 

G 

771 

705 

711 

717 

722 

728 

734 

739 

745 

750 

756 

6 

772 

762 

767 

773 

779 

784 

790 

795 

801 

807 

812 

6 

773 

818 

824 

829 

835 

840 

846 

852 

857 

863 

868 

6 

774 

874 

880 

885 

891 

897 

902 

908 

913 

919 

925 

6 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

1G 


775-819 


LOGAKITHMS. 


88939-91376 


I, 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

775 

88930 

936 

941 

947 

953 

958 

964 

969 

975 

981 

6 

776 

986 

992 

997 

o03 

o09 

o!4 

o20 

o25 

o31 

o37 

6 

777 

89042 

048 

053 

059 

064 

070 

076 

081 

087 

092 

6 

778 

098 

104 

109 

115 

120 

126 

131 

137 

143 

148 

6 

779 

154 

159 

165 

170 

176 

182 

187 

193 

198 

204 

6 

780 

209 

215 

221 

226 

232 

237 

243 

248 

254 

260 

6 

781 

265 

271 

276 

282 

287 

293 

298 

304 

310 

315 

6 

782 

321 

326 

332 

337 

343 

348 

354 

360 

365 

371 

6 

783 

376 

382 

387 

393 

398 

404 

409 

415 

421 

426 

6 

784 

432 

437 

443 

448 

454 

459 

465 

470 

476 

481 

6 

785 

487 

492 

498 

504 

509 

515 

520 

526 

531 

537 

6 

786 

542 

548 

553 

559 

564 

570 

575 

581 

586 

592 

6 

787 

597 

603 

609 

614 

620 

625 

631 

636 

642 

647 

6 

788 

653 

658 

664 

669 

675 

680 

686 

691 

697 

702 

6 

789 

708 

713 

719 

724 

730 

735 

741 

746 

752 

757 

6 

790 

763 

768 

774 

779 

785 

790 

796 

801 

807 

812 

5 

791 

818 

823 

829 

834 

840 

845 

851 

856 

862 

867 

5 

792 

873 

878 

883 

889 

894 

900 

905 

911 

916 

922 

5 

793 

927 

933 

938 

944 

949 

955 

960 

966 

971 

977 

5 

794 

982 

988 

993 

998 

o04 

o09 

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o20 

o26 

o31 

5 

795 

90037 

042 

048 

053 

059 

064 

069 

075 

080 

086 

5 

796 

091 

097 

102 

108 

113 

119 

124 

129 

135 

140 

5 

797 

146 

151 

157 

162 

168 

173 

179 

184 

189 

195 

5 

798 

200 

206 

211 

217 

222 

227 

233 

238 

244 

249 

5 

799 

255 

260 

266 

271 

276 

282 

287 

293 

298 

304 

5 

800 

309 

314 

320 

325 

331 

336 

342 

347 

352 

358 

5 

801 

363 

369 

374 

380 

385 

390 

396 

401 

407 

412 

5 

802 

417 

423 

428 

434 

439 

445 

450 

455 

461 

466 

5 

803 

472 

477 

482 

488 

493 

499 

504 

509 

515 

520 

5 

804 

526 

531 

536 

542 

547 

553 

558 

563 

569 

574 

5 

805 

580 

585 

590 

596 

601 

607 

612 

617 

623 

628 

5 

806 

634 

639 

644 

650 

655 

660 

666 

671 

677 

682 

5 

807 

687 

693 

698 

703 

709 

714 

720 

725 

730 

736 

5 

808 

741 

747 

752 

757 

763 

768 

773 

779 

784 

789 

5 

809 

795 

800 

806 

811 

816 

822 

827 

832 

838 

843 

5 

810 

849 

854 

859 

865 

870 

875 

881 

886 

891 

897 

5 

811 

902 

907 

913 

918 

924 

929 

934 

940 

945 

950 

5 

812 

956 

961 

966 

972 

977 

982 

988 

993 

998 

o04 

5 

813 

91009 

014 

020 

025 

030 

036 

041 

046 

052 

057 

5 

814 

062 

068 

073 

078 

084 

089 

094 

100 

105 

110 

5 

815 

116 

121 

126 

132 

137 

142 

148 

153 

158 

164 

5 

816 

169 

174 

180 

185 

190 

196 

201 

206 

212 

217 

5 

817 

222 

228 

233 

238 

243 

249 

254 

259 

265 

270. 

5 

818 

275 

281 

286 

291 

297 

302 

307 

312 

318 

323 

5 

819 

328 

334 

339 

344 

350 

355 

360 

365 

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376 

5 

I, 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D, 

17 


820-864. 


LOGARITHMS. 


91381-93697. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D, 

820 

91381 

387 

392 

397 

403 

408 

413 

418 

424 

429 

5 

821 

434 

440 

445 

450 

455 

461 

466 

471 

477 

482 

5 

822 

487 

492 

498 

503 

508 

514 

519 

524 

529 

535 

5 

823 

540 

545 

551 

556 

561 

566 

572 

577 

582 

587 

5 

824 

593 

598 

603 

609 

614 

619 

624 

630 

635 

640 

5 

825 

645 

651 

656 

661 

666 

672 

677 

682 

687 

693 

5 

826 

698 

703 

709 

714 

719 

724 

730 

735 

740 

745 

5 

827 

751 

756 

761 

766 

772 

777 

782 

787 

793 

798 

5 

828 

803 

808 

814 

819 

824 

829 

834 

840 

845 

850 

5 

829 

855 

861 

866 

871 

876 

882 

887 

892 

897 

903 

5 

830 

908 

913 

918 

924 

929 

934 

939 

944 

950 

955 

5 

831 

960 

965 

971 

976 

981 

986 

991 

997 

o02 

o07 

5 

832 

92012 

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023 

028 

033 

038 

044 

049 

054 

059 

5 

833 

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070 

075 

080 

085 

091 

096 

101 

106 

111 

5 

834 

117 

122 

127 

132 

137 

143 

148 

153 

158 

163 

5 

835 

169 

174 

179 

184 

189 

195 

200 

205 

210 

215 

5 

836 

221 

226 

231 

236 

241 

247 

252 

257 

262 

267 

5 

837 

273 

278 

283 

288 

293 

298 

304 

309 

314 

319 

5 

838 

324 

330 

335 

340 

345 

350 

355 

361 

366 

371 

5 

839 

376 

381 

387 

392 

397 

402 

407 

412 

418 

423 

5 

840 

428 

433 

438 

443 

449 

454 

459 

464 

469 

474 

5 

841 

480 

485 

490 

495 

500 

505 

511 

516 

521 

526 

5 

842 

531 

536 

542 

547 

552 

557 

562 

567 

572 

578 

5 

843 

583 

588 

593 

598 

603 

609 

614 

619 

624 

629 

5 

844 

634 

639 

645 

650 

655 

.  660 

665 

670 

675 

681 

5 

845 

686 

691 

696 

701 

706 

711 

716 

722 

727 

732 

5 

846 

737 

742 

747 

752 

758 

763 

768 

773 

778 

783 

5 

847 

788 

793 

799 

804 

809 

814 

819 

824 

829 

834 

5 

848 

840 

845 

850 

855 

860 

865 

870 

875 

881 

886 

5 

849 

891 

896 

901 

906 

911 

916 

921 

927 

932 

937 

5 

850 

942 

947 

952 

957 

962 

967 

973 

978 

983 

988 

5 

851 

993 

998 

o03 

o08 

o!3 

o!8 

o24 

o29 

o34 

o39 

5 

852 

93044 

049 

054 

059 

064 

069 

075 

080 

085 

090 

5 

853 

095 

100 

105 

110 

115 

120 

125 

131 

136 

141 

5 

854 

146 

151 

156 

161 

166 

171 

176 

181 

186 

192 

5 

855 

197 

202 

207 

2J2 

217 

222 

227 

232 

237 

242 

5 

856 

247 

252 

258 

263 

268 

273 

278 

283 

288 

293 

5 

857 

298 

303 

308 

313 

318 

323 

328 

334 

339 

344 

5 

858 

349 

354 

359 

364 

369 

374 

379 

384 

389 

394 

5 

859 

399 

404 

409 

414 

420 

425 

430 

435 

440 

445 

5 

860 

450 

455 

460 

465 

470 

475 

480 

485 

490 

495 

5 

861 

500 

505 

510 

515 

520 

526 

531 

536 

541 

546 

5 

862 

551 

556 

561 

566 

571 

576 

581 

586 

591 

596 

5 

863 

601 

606 

611 

616 

621 

626 

631 

636 

641 

646 

5 

864 

651 

656 

661 

666 

671 

676 

682 

687 

692 

697 

5 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

18 


865-909 


LOGARITHMS. 


93702-95899 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D, 

865 

93702 

707 

712 

717 

722 

727 

732 

737 

742 

747 

5 

866 

752 

757 

762 

767 

772 

777 

782 

787 

792 

797 

5 

867 

802 

807 

812 

817 

822 

827 

832 

837 

842 

847 

5 

868 

852 

857 

862 

867 

872 

877 

882 

887 

892 

897 

5 

869 

902 

907 

912 

917 

922 

927 

932 

937 

942 

947 

5 

870 

952 

957 

962 

967 

972 

977 

982 

987 

992 

997 

5 

871 

94002 

007 

012 

017 

022 

027 

032 

037 

042 

047 

5 

872 

052 

057 

062 

067 

072 

077 

082 

086 

091 

096 

5 

873 

101 

106 

111 

116 

121 

126 

131 

136 

141 

146 

5 

874 

151 

156 

161 

166 

171 

176 

181 

186 

191 

196 

5 

875 

201 

206 

211 

216 

221 

226 

231 

236 

240 

245 

5 

876 

250 

255 

260 

265 

270 

275 

280 

285 

290 

295 

5 

877 

300 

305 

310 

315 

320 

325 

330 

335 

340 

345 

5 

878 

349 

354 

359 

364 

369 

374 

379 

384 

389 

394 

5 

879 

399 

404 

409 

414 

419 

424 

429 

433 

438 

443 

5 

880 

448 

453 

458 

463 

468 

473 

478 

483 

488 

493 

5 

881 

498 

503 

507 

512 

517 

522 

527 

532 

537 

542 

5 

882 

547 

552 

557 

562 

567 

571 

576 

581 

586 

591 

5 

883 

596 

601 

606 

611 

616 

621 

626 

630 

635 

640 

5 

884 

645 

650 

655 

660 

665 

670 

675 

680 

685 

689 

5 

885 

694 

699 

704 

709 

714 

719 

724 

729 

734 

738 

5 

886 

743 

748 

753 

758 

763 

768 

T73 

778 

783 

787 

5 

887 

792 

797 

802 

807 

812 

817 

822 

827 

832 

836 

5 

888 

841 

846 

851 

856 

861 

866 

871 

876 

880 

885 

5 

889 

890 

895 

900 

905 

910 

915 

919 

924 

929 

934 

5 

890 

939 

944 

949 

954 

959 

963 

968 

973 

978 

983 

5 

891 

988 

993 

998 

o02 

o07 

o!2 

o!7 

o22 

o27 

o32 

5 

892 

95036 

041 

046 

051 

056 

061 

066 

071 

075 

080 

5 

893 

085 

090 

095 

100 

105 

109 

114 

119 

124 

129 

5 

894 

134 

139 

143 

148 

153 

158 

163 

168 

173 

177 

5 

895 

182 

187 

192 

197 

202 

207 

211 

216 

221 

226 

5 

896 

231 

236 

240 

245 

250 

255 

260 

265 

270 

274 

5 

897 

279 

284 

289 

294 

299 

303 

308 

313 

318 

323 

5 

898 

328 

332 

337 

342 

347 

352 

357 

361 

366 

371 

5 

899 

376 

381 

386 

390 

395 

400 

405 

410 

415 

419 

5 

900  • 

424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

5 

901 

472 

477 

482 

487 

492 

497 

501 

506 

511 

516 

5 

902 

521 

525 

530 

535 

540 

545 

550 

554 

559 

564 

5 

903 

569 

574 

578 

583 

588 

593 

598 

602 

607 

612 

5 

904 

617 

622 

626 

631 

636 

641 

646 

650 

655 

660 

5 

905 

'  665 

670 

674 

679 

684 

689 

694 

698 

703 

708 

5 

906 

713 

718 

722 

727 

732 

737 

742 

746 

751 

756 

5 

907 

761 

766 

770 

775 

7SO 

785 

789 

794 

799 

804 

5 

908 

809 

813 

818 

823 

828 

832 

837 

842 

847 

852 

5 

909 

856 

861 

866 

871 

875 

880 

885 

890 

895 

899 

5 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D, 

19 


910-954 


LOGARITHMS. 


95904-97996. 


ft 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

910 

95904 

909 

914 

918 

923 

928 

933 

938 

942 

947 

5 

911 

952 

957 

961 

966 

971 

976 

980 

985 

990 

995 

5 

912 

999 

o04 

o09 

o!4 

o!9 

o23 

o28 

o33 

o38 

o42 

5 

913 

96047 

052 

057 

061 

066 

071 

076 

080 

085 

090 

5 

914 

095 

099 

104 

109 

114 

118 

123 

128 

133 

137 

5 

915 

142 

147 

152 

156 

161 

166 

171 

175 

180 

185 

5 

916 

190 

194 

199 

204 

209 

213 

218 

223 

227 

232 

5 

917 

237 

242 

246 

251 

256 

261 

265 

270 

275 

280 

5 

918 

284 

289 

294 

298 

303 

308 

313 

317 

322 

327 

5 

919 

332 

336 

341 

346 

350 

355 

360 

365 

369 

374 

5 

920 

379 

384 

388 

393 

398 

402 

407 

412 

417 

421 

5 

921 

426 

431 

435 

440 

445 

450 

454 

459 

464 

468 

5 

922 

473 

478 

483 

487 

492 

497 

501 

506 

511 

515 

5 

923 

520 

525 

530 

534 

539 

544 

548 

553 

558 

562 

5 

924 

567 

572 

577 

581 

586 

591 

595 

600 

605 

609 

5 

925 

614 

619 

624 

628 

633 

638 

642 

647 

652 

656 

5 

926 

661 

666 

670 

675 

680 

685 

689 

694 

699 

703 

5 

927 

708 

713 

717 

722 

727 

731 

736 

741 

745 

750 

5 

928 

755 

759 

764 

769 

774 

778 

783 

788 

792 

797 

5 

929 

802 

806 

811 

816 

820 

825 

830 

834 

839 

844 

5 

930 

848 

853 

858 

862 

867 

872 

876 

881 

886 

890 

5 

931 

895 

900 

904 

909 

914 

918 

923 

928 

932 

937 

5 

932 

942 

946 

951 

956 

960 

965 

970 

974 

979 

984 

5 

933 

988 

993 

997 

o02 

o07 

oil 

o!6 

o21 

o25 

o30 

5 

934 

97035 

039 

044 

049 

053 

058 

063 

067 

072 

077 

5 

935 

081 

086 

090 

095 

100 

104 

109 

114 

118 

123 

5 

936 

128 

132 

137 

142 

146 

151 

155 

160 

165 

169 

5 

937 

174 

179 

183 

188 

192 

197 

202 

206 

211 

216 

5 

938 

220 

225 

230 

234 

239 

243 

248 

253 

257 

262 

5 

939 

267 

271 

276 

280 

285 

290 

294 

299 

.304 

308 

5 

940 

313 

317 

322 

327 

331 

336 

340 

345 

350 

354 

5 

941 

359 

364 

368 

373 

377 

382 

387 

391 

396 

400 

5 

942 

405 

410 

414 

419 

424 

428 

433 

437 

442 

447 

5 

943 

451 

456 

460 

465 

470 

474 

479 

483 

488 

493 

5 

944 

497 

502 

506 

511 

516 

520 

525 

529 

534 

539 

5 

945 

543 

548 

552 

557 

562 

566 

571 

575 

580 

585 

5 

946 

589 

594 

598 

603 

607 

612 

617 

621 

626 

630 

5 

947 

635 

640 

644 

649 

653 

658 

663 

667 

672 

676 

5 

948 

681 

685 

690 

695 

699 

704 

708 

713 

717 

722 

5 

949 

727- 

731 

736 

740 

745 

749 

754 

759 

763 

768 

5 

950 

772 

777 

782 

786 

791 

795 

800 

804 

809 

813 

5 

951 

818 

823 

827 

832 

836 

841 

845 

850 

855 

859 

5 

952 

864 

868 

873 

877 

882 

886 

891 

896 

900 

905 

5 

953 

909 

914 

918 

923 

928 

932 

937 

941 

946 

950 

5 

954 

955 

959 

964 

968 

973 

978 

982 

987 

991 

996 

5 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D, 

20 


955-999 


LOGARITHMS. 


98000-99996 


N, 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

955 

98000 

005 

009 

014 

019 

023 

028 

032 

037 

041 

5 

956 

046 

050 

055 

059 

064 

068 

073 

078 

082 

087 

•  5 

957 

091 

096 

100 

105 

109 

114 

118 

123 

J27 

132 

5 

958 

137 

141 

146 

150 

155 

159 

164 

168 

173 

177 

5 

959 

182 

186 

191 

195 

200 

204 

209 

214 

218 

223 

5 

960 

227 

232 

236 

241 

245 

250 

254 

259 

263 

268 

5 

961 

272 

277 

281 

286 

290 

295' 

299 

304 

308 

313 

5 

962 

318 

322 

327 

331 

336 

340 

345 

349 

354 

358 

5 

963 

363 

367 

372 

376 

381 

385 

390 

394 

399 

403 

5 

964 

408 

4J2 

417 

421 

426 

430 

435 

439 

444 

448 

5 

965 

453 

457 

462 

466 

471 

475 

480 

484 

489 

493 

4 

966 

498 

502 

507 

511 

516 

520 

525 

529 

534 

538 

4 

967 

543 

547 

552 

556 

561 

565 

570 

574 

579 

583 

4 

968 

588 

592 

597 

601 

605 

610 

614 

619 

623 

628 

4 

969 

632 

637 

641 

646 

650 

655 

659 

664 

668 

673 

4 

970 

677 

682 

686 

691 

695 

700 

704 

709 

713 

717 

4 

971 

722 

726 

731 

735 

740 

744 

749 

753 

758 

762 

4 

972 

767 

771 

776 

780 

784 

789 

793 

798 

802 

807 

4 

973 

811 

816 

820 

825 

829 

834 

838 

843 

847 

851 

4 

974 

856 

860 

865 

869 

874 

878 

883 

887 

892 

896 

4 

975 

900 

905 

909 

914 

918 

923 

927 

932 

936 

941 

4 

976 

945 

949 

954 

958 

963- 

967 

972 

976 

981 

985 

4 

977 

989 

994 

998 

o03 

o07 

o!2 

o!6 

o21 

o25 

o29 

4 

978 

99034 

038 

043 

047 

052 

056 

061 

065 

069 

074 

4 

979 

078 

083 

087 

092 

096 

100 

105 

109 

114 

118 

4 

980 

123 

127 

131 

136 

140 

145 

149 

154 

158 

162 

4 

981 

167 

171 

176 

180 

185 

189 

193 

198 

202 

207 

4 

982 

211 

216 

220 

224 

229 

233 

238 

242 

247 

251 

4 

983 

255 

260 

264 

269 

273 

277 

282 

286 

291 

295 

4 

984 

300 

304 

308 

313 

317 

322 

326 

330 

335 

339 

4 

985 

344 

348 

352 

357 

361 

366 

370 

374 

379 

383 

4 

986 

388 

392 

396 

401 

405 

410 

414 

419 

423 

427 

4 

987 

432 

436 

441 

445 

449 

454 

458 

463 

467 

471 

4 

988 

476 

480 

484 

489 

493 

498 

502 

506 

511 

515 

4 

989 

520 

524 

528 

533 

537 

542 

546 

550 

555 

559 

4 

990 

564 

568 

572 

577 

581 

585 

590 

594 

599 

603 

4 

991 

607 

612 

616 

621 

625 

629 

634 

638 

642 

647 

4 

992 

651 

656 

660 

664 

669 

673 

977 

682 

686 

691 

4 

993 

695 

699 

704 

708 

712 

717 

721 

726 

730 

734 

4 

994 

739 

743 

747 

752 

756 

760 

765 

769 

774 

778 

4 

995 

782 

787 

791 

795 

800 

804 

808 

813 

817 

822 

4 

996 

826 

830 

835 

839 

843 

848 

852 

856 

861 

865 

4 

997 

870 

874 

878 

883 

887 

891 

896 

900 

904 

909 

4 

998 

913 

917 

922 

926 

930 

935 

939 

944 

948 

952 

4 

999 

957 

961 

965 

970 

974 

978 

983 

987 

991 

996 

4 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

1000-1044 


LOGARITHMS. 


00000-01907 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

1000 

00000 

004 

009 

013 

017 

022 

026 

030 

035 

039 

4 

1001 

043 

048 

052 

056 

061 

065 

069 

074 

078 

082 

4 

1002 

087 

091 

095 

100 

104 

108 

113 

117 

121 

126 

4 

1003 

130 

134 

139 

143 

147 

152 

156 

160 

165 

169 

4 

1004 

173 

178 

182 

186 

191 

195 

199 

204 

208 

212 

4 

1005 

217 

221 

225 

230 

234 

238 

243 

247 

251 

255 

4 

1006 

260 

264 

268 

273 

277 

281 

286 

290 

294 

299 

4 

1007 

303 

307 

312 

316 

320 

325 

329 

333 

337 

342 

4 

1008 

346 

350 

355 

359 

363 

368 

372 

376 

381 

385 

4 

1009 

389 

393 

398 

402 

406 

411 

415 

419 

424 

428 

4 

1010 

432 

436 

441 

445 

449 

454 

458 

462 

467 

471 

4 

1011 

475 

479 

484 

488 

492 

497 

501 

505 

509 

514 

4 

1012 

518 

522 

527 

531 

535 

540 

544 

548 

552 

557 

4 

1013 

561 

565 

570 

574 

578 

582 

587 

591 

595 

600 

* 

1014 

604 

608 

612 

617 

621 

625 

629 

634 

638 

642 

4 

1015 

647 

651 

655 

659 

664 

668 

672 

677 

681 

685 

4 

1016 

689 

694 

698 

702 

706 

711 

715 

719 

724 

728 

4 

1017 

732 

736 

741 

745 

749 

753 

758 

762 

766 

771 

4 

1018 

775 

779 

783 

788 

792 

796 

800 

805 

809 

813 

4 

1019 

817 

822 

826 

830 

834 

839 

843 

847 

852 

856 

4 

1020 

860 

864 

869 

873 

877 

881 

886 

890 

894 

898 

4 

1021 

903 

907 

911 

915 

920 

924 

928 

932 

937 

941 

4 

1022 

945 

949 

954 

958 

962 

966 

971 

975 

979 

983 

4 

1023 

988 

992 

996 

oOO 

o05 

o09 

o!3 

o!7 

o22 

o26 

4 

1024 

01030 

034 

038 

043 

047 

051 

055 

060 

064 

068 

4 

1025 

072 

077 

081 

085 

089 

094 

098 

102 

106 

111 

4 

1026 

115 

119 

123 

127 

132 

136 

140 

144 

149 

153 

4 

1027 

157 

161 

166 

170 

174 

178 

182 

187 

191 

195 

4 

1028 

199 

204 

208 

212 

216 

220 

225 

229 

233 

237 

4 

1029 

242 

246 

250 

254 

258 

263 

267 

271 

275 

280 

4 

1030 

284 

288 

292 

296 

301 

305 

309 

313 

317 

322 

4 

1031 

326 

330 

334 

339 

343 

347 

351 

355 

360 

364 

4 

1032 

368 

372 

376 

381 

385 

389 

393 

397 

402 

406 

4 

1033 

410 

414 

418 

423 

427 

431 

435 

439 

444 

448 

4 

1034 

452 

456 

460 

465 

469 

473 

477 

481 

486 

490 

4 

1035 

494 

498 

502 

507 

511 

515 

519 

523 

528 

532 

4 

1036 

536 

540 

544 

549 

553 

557 

561 

565 

569 

574 

4 

1037 

578 

582 

586 

590 

595 

599 

603 

607 

611 

616 

4 

1038 

620 

624 

628 

632 

636 

641 

645 

649 

653 

657 

4 

1039 

662 

666 

670 

674 

678 

682 

687 

691 

695 

699 

4 

1040 

703 

708 

712 

716 

720 

724 

728 

733 

737 

741 

4 

1041 

745 

749 

753 

758 

762 

766 

770 

774 

778 

783 

4 

1042 

787 

791 

795 

799 

803 

808 

812 

816 

820 

824 

4 

1043 

828 

833 

837 

841 

845 

849 

853 

868 

862 

866 

4 

1044 

870 

874 

878 

883 

887 

891 

895 

899 

903 

907 

4 

N, 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

1045-1089 


LOGARITHMS. 


01912-03739 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

1045 

01912 

916 

920 

(,»24 

928 

932 

937 

941 

945 

949 

4 

1046 

953 

957 

961 

%6 

970 

974 

978 

982 

986 

991 

4 

1047 

995 

999 

o03 

o07 

oil 

o!5 

o20 

o24 

o28 

o32 

4 

1048 

02036 

040 

044 

049 

053 

057 

061 

065 

069 

073 

4 

1049 

078 

082 

086 

090 

094 

098 

102 

107 

111 

115 

4 

1050 

119 

123 

127 

131 

135 

140 

144 

148 

152 

156 

4 

1051 

160 

164 

169 

173 

177 

181 

185 

189 

193 

197 

4 

1052 

202 

206 

210 

214 

218 

222 

226 

230 

235 

239 

4 

1053 

243 

247 

251 

255 

259 

263 

268 

272 

276 

280 

4 

1054 

284 

288 

292 

296 

301 

305 

309 

313 

317 

321 

4 

1055 

325 

329 

333 

338 

342 

346 

350 

354 

358 

362 

4 

1056 

366 

371 

375 

379 

383 

387 

391 

395 

399 

403 

4 

1057 

407 

412 

416 

420 

424 

428 

432 

436 

440 

444 

4 

1058 

449 

453 

457 

461 

465 

469 

473 

477 

481 

485 

4 

1059 

490 

494 

498 

502 

506 

510 

514 

518 

522 

526 

4 

1060 

531 

535 

539 

543 

547 

551 

555 

559 

563 

567 

4 

1061 

572 

576 

580 

584 

588 

592 

596 

600 

604 

608 

4 

1062 

612 

617 

621 

625 

629 

633 

637 

641 

645 

649 

4 

1063 

653 

657 

661 

666 

670 

674 

678 

682 

686 

690 

4 

1064 

694 

698 

702 

706 

710 

715 

719 

723 

727 

731 

4 

1065 

735 

739 

743 

747 

751 

755 

759 

763 

768 

772 

4 

1066 

776 

780 

784 

788 

792 

796 

800 

804 

808 

812 

4 

1067 

816 

821 

825 

829 

833 

837 

841 

845 

849 

853 

4 

1068 

857 

861 

865 

869 

873 

877 

882 

886 

890 

894 

4 

1069 

898 

902 

906 

910 

914 

918 

922 

926 

930 

934 

4 

1070 

938 

942 

946 

951 

955 

959 

963 

967 

971 

975 

4 

1071 

979 

983 

987 

991 

995 

999 

o03 

o07 

oil 

o!5 

4 

1072 

03019 

024 

028 

032 

036 

040 

044 

048 

052 

056 

4 

1073 

060 

064 

068 

072 

076 

080 

084 

088 

092 

096 

4 

1074 

100 

104 

109 

113 

117 

121 

125 

129 

133 

137 

4 

1075 

141 

145 

149 

153 

157 

161 

165 

169 

173 

177 

4 

1076 

181 

185 

189 

193 

197 

201 

205 

209 

214 

218 

4 

1077 

222 

226 

230 

234 

238 

242 

246 

250 

254 

258 

4 

1078 

262 

266 

270 

274 

278 

282 

286 

290 

294 

298 

4 

1079 

302 

306 

310 

314 

318 

322 

326 

330 

334 

338 

4 

1080 

342 

346 

350 

354 

358 

362 

366 

371 

375 

379 

4 

1081 

383 

387 

391 

395 

399 

403 

407 

411 

415 

419 

4 

1082 

423 

427 

431 

435 

439 

443 

447 

451 

455 

459 

4 

1083 

463 

467 

471 

475 

479 

483 

487 

491 

495 

499 

4 

1084 

503 

507 

511 

515 

519 

523 

527 

531 

535 

539 

4 

1085 

543 

547 

551 

555 

559 

563 

567 

571 

575 

579 

4 

1086 

583 

587 

591 

595 

599 

603 

607 

611 

615 

619 

4 

1087 

623 

627 

631 

635 

639 

643 

647 

651 

655 

659 

4 

1088 

663 

667 

671 

675 

679 

683 

687 

691 

695 

699 

4 

1089 

703 

707 

711 

715 

719 

723 

727 

731 

735 

739 

4 

TS. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D, 

23 


II.  NATURAL  SINES. 


Deg, 

0' 

10' 

20' 

30! 

40' 

50' 

Deg, 

0 

00000 

00291 

00582 

00873 

01164 

01454 

01745 

89 

1 

01745 

02036 

02327 

02618 

02908 

03199 

03490 

88 

2 

03490 

03781 

04071 

04362 

04653 

04943 

05234 

87 

3 

05234 

05524 

05814 

06105 

06395 

06685 

06976 

86 

4 

06976 

07266 

07556 

07846 

08136 

08426 

08716 

85 

5 

08716 

09005 

09295 

09585 

09874 

10164 

10453 

84 

6 

10453 

10742 

11031 

11320 

11609 

11898 

12187 

83  * 

7 

12187 

12476 

12764 

13053 

13341 

13629 

13917 

82 

8 

13917 

14205 

14493 

14781 

15069 

15356 

15643 

81 

9 

15643 

15931 

16218 

16505 

16792 

17078 

17365 

80 

10 

17365 

17651 

17937 

18224 

18509 

18795 

19081 

79 

11 

19081 

19366 

19652 

19937 

20222 

20507 

20791 

78 

12 

20791 

21076 

21360 

21644 

21928 

22212 

22495 

77 

13 

22495 

22778 

23062 

23345 

23627 

23910 

24192 

76 

14 

24192 

24474 

24756 

25038 

25320 

25601 

25882 

75 

15 

25882 

26163 

26443 

26724 

27004 

27284 

27564 

74 

16 

27564 

27843 

28123 

28402 

28680 

28959 

29237 

73 

17 

29237 

29515 

29793 

30071 

30348 

30625 

30902 

72 

18 

30902 

31178 

31454 

31730 

32006 

32282 

32557 

71 

19 

32557 

32832 

33106 

33381 

33655 

33929 

34202 

70 

20 

34202 

34475 

34748 

35021 

35293 

35565 

35837 

69 

21 

35837 

36108 

36379 

36650 

36921 

37191 

37461 

68 

22 

37461 

37730 

37999 

38268 

38537 

38805 

39073 

67 

23 

39073 

39341 

39608 

39875 

40141 

40408 

40674 

66 

24 

40674 

40939 

41204 

41469 

41734 

41998 

42262 

65 

25 

42262 

42525 

42788 

43051 

43313 

43575 

43837 

64 

26 

43837 

44098 

44359 

44620 

44880 

45140 

45399 

63 

27 

45399 

45658 

45917 

46175 

46433 

46690 

46947 

62 

28 

46947 

47204 

47460 

47716 

47971 

48226 

48481 

61 

29 

48481 

48735 

48989 

49242 

49495 

49748 

50000 

60 

30 

50000 

50252 

50503 

50754 

51004 

51254 

51504 

59 

31 

51504 

51753 

52002 

52250 

52498 

52745 

52992 

58 

32 

52992 

53238 

53484 

53730 

53975 

54220 

54464 

57 

33 

54464 

54708 

54951 

55194 

55436 

55678 

55919 

56 

34 

55919 

56160 

56401 

56641 

56880 

57119 

57358 

55 

35 

57358 

57596 

57833 

58070 

58307 

58543 

58779 

54 

36 

58779 

59014 

59248 

59482 

59716 

59949 

60182 

53 

37 

60182 

60414 

60645 

60876 

61107 

61337 

61566 

52 

38 

61566 

61795 

62024 

62251 

62479 

62706 

62932 

51 

39 

62932 

63158 

63383 

63608- 

63832 

64056 

64279 

50 

40 

64279 

64501 

64723 

64945 

65166 

65386 

65606 

49 

41 

65606 

65825 

66044 

66262 

66480 

66697 

66913 

48 

42 

66913 

67129 

67344 

67559 

67773 

67987 

68200 

47 

43 

68200 

68412 

68624 

68835 

69046 

69256 

69466 

46 

44 

69466 

69675 

69883 

70091 

70298 

70505 

70711 

45 

Deg, 

5<y 

4(X 

3(y 

20' 

10' 

(X 

.Deg, 

NATURAL  COSINES. 


II.  NATURAL   SINES. 


Deg, 

0' 

10' 

20' 

30' 

40' 

50' 

Deg- 

45 

70711 

70916 

71121 

71325 

71529 

71732 

71934 

44 

46 

71934 

72136 

72337 

72537 

72737 

72937 

73135 

43 

47 

73135 

73333 

73531 

73728 

73924 

74120 

74314 

42 

48 

74314 

74509 

74703 

74896 

75088 

75280 

75471 

41 

49 

75471 

75661 

75851 

76041 

76229 

76417 

76604 

40 

50 

76604 

76791 

76977 

77162 

77347 

77531 

77715 

39 

51 

77715 

77897 

78079 

78261 

78442 

78622 

78801 

38 

52 

78801 

78980 

79158 

79335 

79512 

79688 

79864 

37 

53 

79864 

80038 

80212 

80386 

80558 

80730 

80902 

36 

54 

80902 

81072 

81242 

81412 

81580 

81748 

81915 

35 

55 

81915 

82082 

82248 

82413 

82577 

82741 

82904 

34 

56 

82904 

83066 

83228 

83389 

83549 

83708 

83867 

33 

57 

83867 

84025 

84182 

84339 

84495 

84650 

84805 

32 

58 

84805 

84959 

85112 

85264 

85416 

85567 

85717 

31 

59 

85717 

85866 

86015 

86163 

86310 

86457 

86603 

30 

60 

86603 

86748 

86892 

87036 

87178 

87321 

87462 

29 

61 

87462 

87603 

87743 

87882 

88020 

88158 

88295 

28 

62 

88295 

88431 

88566 

88701 

88835 

88968 

89101 

27 

63 

89101 

89232 

89363 

89493 

89623 

89752 

89879 

26 

64 

89879 

90007 

90133 

90259 

90383 

90507 

90631 

25 

65 

90631 

90753 

90875 

90996 

91116 

91236 

91355 

24 

66 

91355 

91472 

91590 

91706 

91822 

91936 

92050 

23 

67 

92050 

92164 

92276 

92388 

92499 

92609 

92718 

22 

68 

92718 

92827 

92935 

93042 

93148 

93253 

93358 

21 

69 

93358 

93462 

93565 

93667 

93769 

93869 

93969 

20 

70 

93969 

94068' 

94167 

94264 

94361 

94457 

94552 

19 

71 

94552 

94646 

94740 

94832 

94924 

95015 

95106 

18 

72 

95106 

95195 

95284 

95372 

95459 

95545 

95630 

17 

73 

95630 

95715 

95799 

95882 

95964 

96046 

96126 

16 

74 

96126 

96206 

96285 

96363 

96440 

96517 

96593 

15 

75 

96593 

96667 

96742 

96815 

96887 

96959 

97030 

14 

76 

97030 

97100 

97169 

97237 

97304 

97371 

97437 

13 

77 

97437 

97502 

97566 

97630 

97692 

97754 

97815 

12 

78 

97815 

97875 

97934 

97992 

98050 

98107 

98163 

11 

79 

98163 

98218 

98272 

98325 

98378 

98430 

98481 

10 

80 

98481 

98531 

98580 

98629 

98676 

98723 

98769 

9 

81 

98769 

98814 

98858 

98902 

98944 

98986 

99027 

8 

82 

99027 

99067 

99106 

99144 

99182 

99219 

99255 

7 

83 

99255 

99290 

99324 

99357 

99390 

99421 

99452 

6 

84 

99452 

99482 

99511 

99540 

99567 

99594 

99619 

5 

85 

99619 

99644 

99668 

99692 

99714 

99736 

99756 

4 

86 

99756 

99776 

99795 

99813 

99831 

99847 

99863 

3 

87 

99863 

99878 

99892 

99905 

99917 

99929 

99939 

2 

88 

99939 

99949 

99958 

99966 

99973 

99979 

99985 

1 

89 

99985 

99989 

99993 

99996 

99998 

99999 

1.0000 

0 

Deg. 

50' 

40' 

30' 

20' 

10' 

0' 

Deg, 

25 


NATURAL  COSINES. 


Ill— NATURAL  TANGENTS. 


Deg, 

0' 

10' 

20' 

30' 

40' 

50' 

Deg, 

0 

00000 

00291 

00582 

00873 

01164 

01455 

01746 

89 

1 

01746 

02036 

02328 

02619 

02910 

03201 

03492 

88 

2 

03492 

03783 

04075 

04366 

04658 

04949 

05241 

87 

3 

05241 

05533 

05824 

06116 

06408 

06700 

06993 

86 

4 

06993 

07285 

07578 

07870 

08163 

08456 

08749 

85 

5 

08749 

09042 

09335 

09629 

09923 

10216 

10510 

84 

6 

10510 

10805 

11099 

11394 

11688 

11983 

12278 

83 

7 

12278 

12574 

12869 

13165 

13461 

13758 

14054 

82 

8 

14054 

14351 

14648 

14945 

15243 

15540 

15838 

81 

9 

15838 

16137 

16435 

16734 

17033 

17333 

17633 

80 

10 

17633 

17933 

18233 

18534 

18835 

19136 

19438 

79 

11 

19438 

19740 

20042 

20345 

20648 

20952 

21256 

78 

12 

21256 

21560 

21864 

22169 

22475 

22781 

23087 

77 

13 

23087 

23393 

23700 

24008 

24316 

24624 

24933 

76 

14 

24933 

25242 

25552 

25862 

26172 

26483 

26795 

75 

15 

26795 

27107 

27419 

27732 

28046 

28360 

28675 

74 

16 

28675 

28990 

29305 

29621 

29938 

30255 

30573 

73 

17 

30573 

30891 

31210 

31530 

31850 

32171 

32492 

72 

18 

32492 

32814 

33136 

33460 

33783 

34108 

34433 

71 

19 

34433 

34758 

35085 

35412 

35740 

36068 

36397 

70 

20 

36397 

36727 

37057 

37388 

37720 

38053 

38386 

69 

21 

38386 

38721 

39055 

39391 

39727 

40065 

40403 

68 

22 

40403 

40741 

41081 

41421 

41763 

42105 

42447 

67 

23 

42447 

42791 

43136 

43481 

43828 

44175 

44523 

66 

24 

44523 

44872 

45222 

45573 

45924 

46277 

46631 

65 

25 

46631 

46985 

47341 

47698 

48055 

48414 

48773 

64 

26 

48773 

49134 

49495 

49858 

50222 

50587 

50953 

63 

27 

50953 

51319 

51688 

52057 

52427 

52798 

53171 

62 

28 

53171 

53545 

53920 

54296 

54673 

55051 

55431 

61 

29 

55431 

55812 

56194 

56577 

56962 

5734$ 

57735 

60 

30 

57735 

58124 

58513 

58905 

59297 

59691 

60086 

59 

31 

60086 

60483 

60881 

61280 

61681 

62083 

62487 

58 

32 

62487 

62892 

63299 

63707 

64117 

64528 

64941 

57 

33 

64941 

65355 

65771 

66189 

66608 

67028 

67451 

56 

34 

67451 

67875 

68301 

68728 

69157 

69588 

70021 

55 

35 

70021 

70455 

70891 

71329 

71769 

72211 

72654 

54 

36 

72654 

73100 

73547 

73996 

74447 

74900 

75355 

53 

37 

75355 

75812 

76272 

76733 

77196 

77661 

78129 

52 

38 

78129 

78598 

79070 

79544 

80020 

80498 

80978 

51 

39 

80978 

81461 

81946 

82434 

82923 

83415 

83910 

50 

40 

83910 

84407 

84906 

85408 

85912 

86419 

86929 

49 

41 

86929 

87441 

87955 

88473 

88992 

89515 

90040 

48 

42 

90040 

90569 

91099 

91633 

92170 

92709 

93252 

47 

43 

93252 

93797 

94345 

94896 

95451 

96008 

96569 

46 

44 

96569 

97133 

97700 

98270 

98843 

99420 

1.00000 

45 

Deg. 

50' 

4<y 

30' 

20' 

10' 

0' 

Deg. 

NATURAL  COTANGENTS. 


III.— NATURAL  TANGENTS. 


Deg, 

0' 

10' 

20' 

30' 

40' 

50' 

Deg, 

45 

1.00000 

1.00583 

1.01170 

1.01761 

1.02355 

1.02952 

1.03553 

44 

46 

1.03553 

1.04158 

1.04766 

1.05378 

1.05994 

1.06613 

1.07237 

43 

47 

1.07237 

1.07864 

1.08496 

1.09131 

1.09770 

1.10414 

1.11061 

42 

48 

1.11061 

1.11713 

1.12369 

1.13029 

1.13694 

1.14363 

1.15037 

41 

49 

1.15037 

1.15715 

1.16398 

1.17085 

1.17777 

1.18474 

1.19175 

40 

50 

1.19175 

1.19882 

1.20593 

1.21310 

1.22031 

1.22758 

1.23490 

39 

51 

1  .23490 

124227 

1.24969 

1.25717 

1.26471 

1.27230 

1.27994 

38 

52 

1.27994 

1.28764 

1.29541 

1.30323 

1.31110 

1.31904 

1.32704 

37 

53 

1.32704 

1.33511 

1.34323 

1.35142 

1.35968 

1.36800 

1.37638 

36 

54 

1.37638 

1.38484 

1.39336 

1.40195 

1.41061 

1.41934 

1.42815 

35 

55 

1.42815 

1.43703 

1.44598 

1.45501 

1.46411 

1.47330 

1.48256 

34 

56 

1.48256 

1.49190 

1.50133 

1.51084 

1.52043 

1.53010 

1.53987 

33 

57 

1.53987 

1.54972 

1.55966 

1.56969 

1.57981 

1.59002 

1.60033 

32 

58 

1.60033 

1.61074 

1.62125 

1.63185 

1.64256 

1.65337 

1.66428 

31 

59 

1.66428 

1.67530 

1.68643 

1.69766 

1.70901 

1.72047 

1.73205 

30 

60 

1.73205 

1.74375 

1.75556 

1.76749 

1.77955 

1.79174 

1.80405 

29 

61 

1.80405 

1.81649 

1.82906 

1.84177 

1.85462 

1.86760 

1.88073 

28 

62 

1.88073 

1.89400 

1.90741 

1.92098 

1.93470 

1.94858 

1.96261 

27 

63 

1.96261 

1.97680 

1.99116 

2.00569 

2.02039 

2.03526 

2.05030 

26 

64 

2.05030 

2.06553 

2.08094 

2.09654 

2.11233 

2.12832 

2.14451 

25 

65 

2.14451 

2.16090 

2.17749 

2.19430 

2.21132 

2.22857 

2.24604 

24 

66 

2.24604 

2.26374 

2.28167 

2.29984 

2.31826 

2.33693 

2.35585 

23 

67 

2.35585 

2.37504 

2.39449 

2.41421 

2.43422 

2.45451 

2.47509 

22 

68 

2.47509 

2.49597 

2.51715 

2.53865 

2.56046 

2.58261 

2.60509 

21 

69 

2.60509 

2.62791 

2.65109 

2.67462 

2.69853 

2.72281 

2.74748 

20 

70 

2.74748 

2.77254 

2.79802 

2.82391 

2.85023 

2.87700 

2.90421 

19 

71 

2.90421 

2.93189 

2.96004 

2.9886<S 

3.01783 

3.04749 

3.07768 

18 

72 

3.07768 

3.10842 

3.13972 

3.17159 

3.20406 

3.23714 

3.27085 

17 

73 

3.27085 

3.30521 

3.34023 

3.37594 

3.41236 

3.44951 

3.48741 

16 

74 

3.48741 

3.52609 

3.56557 

3.60588 

3.64705 

3.68909 

3.73205 

15 

75 

3.73205 

3.77595 

3.82083 

3.86671 

3.91364 

3.96165 

4.01078 

14 

76 

4.01078 

4.06107 

4.11256 

4.16530 

4.21933 

4.27471 

4.33148 

13 

77 

4.33148 

4.38969 

4.44942 

4.51071 

4.57363 

4.63825 

4.70463 

12 

78 

4.70463 

4.77286 

4.84300 

4.91516 

4.98940 

5.06584 

5.14455 

11 

79 

5.14455 

5.22566 

5.30928 

5.39552 

5.48451 

5.57638 

6.67128 

10 

80 

5.67128 

5.76937 

5.87080 

5.97576 

6.08444 

6.19703 

6.31375 

9 

81 

6.31375 

6.43484 

6.56055 

6.69116 

6.82694 

6.96823 

7.11537 

8 

82 

7.11537 

7.26873 

7.42871 

7.59575 

7.77035 

7.95302 

8.14435 

7 

83 

8.14435  8.34496 

8,55555 

8.77689 

9.00983 

9.2553C 

9.51436 

6 

84 

9.51436  9.78817 

10.0780 

10.3854 

10.7119 

11.0594 

11.4301 

5 

85 

11.4301 

11.8262 

12.2505 

12.7062 

13.1969 

13.7267 

14.3007 

4 

86 

14.3007 

14.9244 

15.6048 

16,3499 

17.1693 

18.0750 

19.0811 

3 

87 

19.0811 

20.2056 

21.4704 

22.9038 

24.5418 

26.4316 

28.6363 

2 

88 

28.6363 

31.2416 

34,3678 

38.1885 

42.9641 

49.1039 

57.2900 

1 

89 

57.2900 

68.7501 

85.9398 

114.589 

171.885 

343.774 

00 

0 

Deg 

50' 

40' 

30' 

20' 

10' 

0' 

Deg, 

27 


NATURAL  COTANGENTS. 


TABLE   IV.— LOGARITHMIC 


31. 

Sine. 

Dl" 

THUS. 

Dl" 

M. 

M. 

Sine.   Dl"  |  Tang. 

Dl"  j  M. 

o 

00 

00 

60 

0 

S.24186^  ft  8.24192 

120  60 

1 

2 
3 

4 
5 
6 

7 

6.46373 
76476 
94085 
7.06579 
16270 
24188 
30882 

502 

293 
208 
162 
132 
112 

6.46373 
76476 
94085 
7.06579 
16270 
24188 
30882 

502 
293 
208 
162 
132 
112 

59 
58 
57 
56 
55 
54 
53 

1 

2 
3 
4 
5 
6 
7 

24903 
25609 
26304 
2t)9b8 
27661 
28324 
28977 

11.8 
11.6 
11.4 
11.2 
11.0 
10.9 

24910 
25616 
26312 
26996 
27669 
28332 
28986 

11.8 
11.6 
11.4 
11.2 
11.0 
10.9 

59 
58 
57 
56 
55 
54 
53 

8 
9 
10 
11 
12 

36682 
41797 
46373 
7.50512 
54291 

96.7 
85.2 
76.3 
69.0 
63.0 

36682 
41797 
46373 
7.50512 
54291 

85.2 
76.3 
69.0 
63.0 
57  9 

52 
51 
50 

49 

48 

8 
9 
10 
11 
12 

29621 
30255 
30879 
8.31495 
32103 

10.6 
10.4 
10.3 
10.1 

29629 
30263 
30888 
8.31505 
32112 

10.7 
10.6 
10.4 
10.3 
10.1 

52 
51 
50 
49 

48 

13 
14 

57767 
60985 

53.6 

57767 

60986 

53.6 

47 

46 

13 
14 

32702 
33292 

9.85 

32711 
33302 

9.85 

47 
46 

15 

16 
17 
18 
19 
20 
21 

63982 
66784 
69417 
71900 
74248 
76475 
7.78594 

49.9 
46.7 
43.9 
41.4 
39.1 
37.1 
35.3 

63982 
667*85 
69418 
71900 
74248 
76476 
7.78595 

49.9 
46.7 
43.9 
41.4 
39.1 
37.1 
35.3 

45 
44 
43 
42 
41 
40 
39 

15 

16 
17 
18 

19 
20 
21 

33875 
34450 
35018 
35578 
36131 
36678 
8.37217 

9.71 
9.59 
9.46 
9.34 
9.22 
9.10 
8.99 

33886 
34461 
35029 
35590 
36143 
36689 
8.37229 

9.72 
9.59 
9.47 
9.35 
9.22 
9.11 
9.00 

45 
44 
43 
42 
41 
40 
39 

22 
23 

80615 
82545 

33.7 
32.2 

80615 
82546 

66.  ( 
32.2 

38 
37 

22 
23 

37750 
38276 

8.77 

37762 
38289 

.00 

8.78 

38 
37 

24 
25 

26 
27 
28 
29 
30 
31 
32 
33 

84393 
86166 
87870 
89509 
91088 
92612 
94084 
7.95508 
96887 
98223 

30.8 
29.5 
28.4 
27.3 
26.3 
25.4 
24.5 
23.7 
23.0 
22.3 

84394 
86167 
87871 
89510 
91089 
92613 
94086 
7.95510 
96889 
98225 

30.8 
29.5 
28.4 
27.3 
26.3 
25.4 
24.5 
23.7 
23.0 
22.3 

36 
35 
34 
33 
32 
31 
30 
29 
28 
7.1 

24 
25 
26 
27 
28 
29 
30 
31 
32 
33 

38796 
39310 
39818 
40320 
40816 
41307 
41792 
8.42272 
42746 
43216 

8.67 
8.56 
8.46 
8.37 
8.27 
8.18 
8.09 
8.00 
7.91 
7.S2 

38S09 
39323 
39832 
40334 
40830 
41321 
41807 
8.42287 
42762 
43232 

8.67 
8.57 
8.47 
8.37 
8.28 
8.18 
8.09 
8.0t) 
7.91 
7.83 

36 
35 
34 
33 
32 
31 
30 
29 
28 
27 

34 
35 
36 
37 
38 
39 
40 

99520 
8.00779 
02002 
03192 
04350 
05478 
06578 

21.6 
21.0 
20.4 
19.8 
19.3 
18.8 
18.3 

99522 
8.00781 
02004 
03194 
04353 
05481 
06581 

21.6 
21.0 
20.4 
19.8 
19.3 
18.8 
18.3 

26 
25 
24 
23 
22 
21 
SO 

34 
35 
36 
37 
38 
39 
40 

43680 
44139 
44594 
45044 
45489 
45930 
46366 

7.74 
7.66 
7.58 
7.50 
7.42 
7.35 
7.27 

43696 
44156 
44611 
45061 
45507 
45948 
46385 

7.75 
7.66 
7.58 
7.50 
7.43 
7.35 
7.28 

26 
25 
24 
23 
22 
21 
20 

41 
42 

8.07650 
08696 

17.9 
17.4 

8.07653 
08700 

17.9 
17.4 

19 

18 

41 
42 

8.46799 
47226 

f'f  J  8.46817 

7ftJ   47245 

7.21 
7.13 

19 
18 

43 
44 
45 
46 
47 
48 
49 
50 
51 

09718 
10717 
11693 
12647 
13581 
14495 
15391 
16268 
8.17128 

17.0 
16.6 
16.3 
15.9 
15.6 
15.2 
14.9 
14.6 
14.3 

09722 
10720 
11696 
12651 
13585 
14500 
15395 
16273 
8.17133 

17.0 
16.6 
16.3 
15.9 
15.6 
15.2 
14.9 
14.6 
14.3 

17 
16 
15 
14 
13 
12 
11 
10 
9 

43 
44 
45 
46 
47 
48 
49 
50 
51 

47650 
48069 
48485 
48896 
49304 
49708 
50108 
50504 
8.50897 

6.99 
6.92 
6.86 
6.79 
6.73 
6.67 
6.61 
6.55 
ft  4Q 

476C9 
48089 
48505 
48917 
49325 
49729 
50130 
50527 
8.50920 

7.00 
6.93 
6.87 
6.80 
6.74 
6.68 
6.62 
6.55 

6P,n 

17 
16 
15 
14 
13 
12 
11 
10 
9 

52 
53 
54 
55 
56 
57 
58 
59 
60 

17971 
18798 
19610 
20407 
21189 
21958 
22713 
23456 
24186 

13.8 
13.5 
13.3 
13.0 
12.8 
12.6 
12.4 
12.2 

17976 
18804 
19616 
20413 
21195 
21964 
22720 
23462 
24192 

13.8 
13.5 
13.3 
13.0 
12.8 
12.6 
12.4 
12.2 

8 
7 
6 
5 
4 
3 
2 
1 
0 

52 
53 
54 
55 

56 
57 
58 
59 
60 

51287 
51673 
52055 
52434 
52810 
53183 
53552 
53919 
54282 

6.43 
6.37 
6.32 
6.26 
6.21 
6.16 
6.11 
6.05 

51310 
51696 
52079 
52459 
52835 
53208 
53578 
53945 
54308 

6.44 
6.38 
6.33 
6.27 
6.22 
6.17 
6.11 
6.06 

8 
7 
6 
5 
4 
3 
2 
1 
0 

M. 

Cosine. 

Dl" 

Cotang. 

Dl" 

M. 

M. 

Cosine. 

Dl" 

Cotang. 

Dl" 

M 

89° 


SINES  AND    TANGENTS. 


M. 

Sine.  |  Dl"   THUS.   Dl" 

M. 

.M     Sim-. 

Dl"   Tang. 

Dl"   M. 

0 

8.54282  „ 

8.54308.fi  0] 

60 

0  18.71881) 

8.71940 

0 

1    54642.?'^ 

54669  r'fw, 

59 

1 

72120  *•]" 

72181 

H?  59 

2 

549991  TO  , 

55027  ;^; 

58 

2 

7285Q3 

72420  r":  58 

3 

55354  ™ 

55382  I?-*} 

57 

3 

72597  |*Jjj 

72659 

•5.yn  ry 

Q  Q<S   h' 

4 
5 

55705|  .'Q? 
56054  1  j?yl 

55734 

56083 

5.82 

^  77 

56 
55 

4 
5 

73069  \™t 

72896 
73132 

o.  yo 
3.93 

0  Q-l 

56 
55 

6 

56400  1 

56429  "'I' 

54 

6 

73303  rXX 

73366 

d«91 

54 

7 

567431  M? 

56/73-12 

53 

7 

73535 

73600 

3.89 

53 

8 

57084 

a.o/ 

571141  Hg 

52 

8 

73767 

3.86 

73832 

3.87 

52 

9 
10 

57421 
57757 

5.63 
5.59 

57452  Z'XJ 
57788  X'?: 

51 

50 

9 
10 

73997 
74226 

3.84 
3.82 

74063 
74292 

3.85 
3.83 

51 
50 

11 

8.58089 

5.54 

8.58121  ™ 

49 

11 

8.74454 

3.80 

8.74521 

3.81 

49 

12 

58419 

5.50 

t*  A  A 

58451  ir;?' 

48 

12 

74680 

3.78 

9  7  £. 

74748 

3.79 

3T7 

48 

13 

58747 

D.4O 

58779  J™1 

47 

13 

74906 

•».7o 

74974 

.77 

47 

14 

59072 

5.42 

59105  r*JJ 

46 

14 

75130 

3.74 

75199 

3.75 

46 

15 

59395 

5.38 

59428  ir",r 

45 

15 

7£i353 

3.72 

75423 

3.73 

45 

16 

59715 

5.34 

59749  .'„, 

44 

16 

75575 

3.70 

75645 

3.71 

44 

17 

18 
19 
20 
21 
22 
23 
24 
25 
26 
27 

60033 
60349 
60662 
60973 
8.61282 
61589 
61894 
62196 
62497 
62795 
63091 

5.30 
5.26 
5.22 
5.19 
5.15 
5.11 
5.08 
5.04 
5.01 
4.97 
4.94 

60068 
60384 
60698 
61009 
8.61319 
61626 
61931 
62234 
62535 
62834 
63131 

5.27 
5.23 
5.19 
5.16 
5.12 
5.08 
5.05 
5.02 
4.98 
4.95 

43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 

17 

18 
19 
20 
21 
22 
23 
24 
25 
26 
27 

75795 
76015 
76234 
76451 
8.76667 
76883 
77097 
77310 
77522 
77733 
77943 

3.68 
3.66 
3.64 
3.62 
3.61 
3.59 
3.57 
3.55 
3.53 
3.52 
3.50 

75867 
76087 
76306 
76525 
8.76742 
76958 
77173 
77387 
77600 
77811 
78022 

3.69 
3.67 
3.65 
3.64 
3.62 
3.60 
3.58 
3.57 
3.55 
3.53 
3.51 

43 
42 
41 

40 
39 
38 
37 
36 
35 
34 
33 

28 
29 
30 
31 
32 
33 
34 
35 

37 
38 
39 
41) 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 

63385 
63678 
63968 
8.64256 
64543 
64827 
65110 
65391 
65670 
65947 
66223 
66497 
66769 
8.67039 
67308 
67575 
67841 
68104 
68367 
68627 
68886 
69144 
69400 
8.69654 
69907 
70159 

4.90 
4.87 
4.84 
4.81 
4.78 
4.74 
4.71 
4.68 
4.65 
4.62 
4.59 
4.56 
4.53 
4.51 
4.48 
4.45 
4.42 
4.40 
4.37 
4.34 
4.32 
4.29 
4.27 
4.24 
4.22 
4.19 
4*1  7 

63426 
63718 
64009 
8.64298 
64585 
64870 
65154 
65435 
65715 
65993 
66269 
66543 
66816 
8.67087 
67356 
67624 
67890 
68154 
68417 
68678 
68938 
69196 
69453 
8.69708 
69962 
70214 

4.91 
4.88 
4.85 
4.82 
4.78 
4.75 
4.72 
4.69 
4.66 
4.63 
4.60 
4.57 
4.54 
4.52 
4.49 
4.46 
4.43 
4.41 
4.38 
4.35 
4.33 
4.30 
4.28 
4.25 
4.23 
4.20 

A  1  0 

32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 

28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 

78152 
78360 
78568 
8.78774 
78979 
79183 
79386 
79588 
79789 
79990 
80189 
80388 
80585 
8.80782 
80978 
81173 
81367 
81560 
81752 
81944 
82134 
82324 
82513 
8.82701 
82888 
83075 

3.48 
3.47 
3.45 
3.43 
3.42 
3.40 
3.39 
3.37 
3.35 
3.34 
3.32 
3.31 
3.29 
3.28 
3.26 
3.25 
3  23 
3.22 
3.20 
3.19 
3.18 
3.16 
3.15 
3.13 
3.12 
3.11 

78232 
78441 
78649 
8.78855 
79061 
79266 
79470 
79673 
79875 
80076 
80277 
80476 
80674 
8.80872 
81068 
81264 
81459 
81653 
81846 
82038 
82230 
82420 
82610 
8.82799 
82987 
83175 

3.50 
3.48 
3.46 
3.45 
3.43 
3.42 
3.40 
3.38 
3.37 
3.35 
3.34 
3.32 
3.31 
3.29 
3.28 
3.26 
3.25 
3.23 
3.22 
3.20 
3.19 
3.18 
3.16 
3.15 
3.14 
3.12 

32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
'  20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8  • 
7 

54 

70409 

/t  1  A 

70465 

4*  lo 
41  *S 

6 

54 

83261 

3.10 

•_»  no 

83361 

3.11 

31  A 

6 

55 

70658  1"'1" 

70714 

•  It) 

4  1  3 

5 

55 

83446 

o.Uo 

83547 

.10 

5 

56 
57 
58 
59 
60 

70905  *•" 
71151  J'i? 

'tt»*1Ioi 
niasfJJ 

71880  4'03 

70962 
71208 
71453 
71697 
71940 

•±.10 

4.11 
4.08 
4.06 
4.04 

4 
3 
2 
1 

0 

56 

57 
58 
59 
60 

83630 
83813 
83996 
84177 
84358 

3.07 
3.06 
3.04 
3.03 
3.02 

83732 
83916 
84100 

84282 
84464 

3.08 
3.07 
3.06 
3.04 
3.03 

4 
3 
2 
1 
0 

M. 

f'osine.  Dl" 

Cotang.j  Dl" 

M. 

M. 

Cosine. 

Dl" 

Ootang.  DJ"  j  M. 

S.  N.  37. 


TABLE    IV.— LOGARITHMIC 


M. 

Bine. 

1)1" 

Tang.   Dl"   31. 

.M.    Sim-.    IM" 

Tans. 

Dl" 

M. 

0 

8.84358 

g  t\  i 

x  84404  3  Q2  60 

0  Is.  94030  „  tn 

8.94195 

CO 

2 

84539 

84718 

O.U  1 

2.99 
2.98 

*  84826  H,) 

59 

58 

1 

2 

94174  *•*" 
94317  J-JJ 

9434«  f^ 
94485  ••}{ 

59 

58 

3 

84897 

85006 

57 

3 

1)4461  f  JJ 

94630 

^.1  1 

57 

4 

85075 

2.97 

2O  A 

85185  Hf 

56 

4 

94603  f  Jj 

94773 

2.40 

20ft 

56 

5 

852.32 

.yo 

85363  I'll 

55 

5 

94746  "•;,". 

94917 

.ov 

55 

6 

85429 

2.94 

85540  fir 

54 

6 

94887  "'"*' 

95060 

2.38 

54 

7 
8 
9 
10 

85(i05 
85780 
85955 
86128 

2.93 
2.92 
2.91 
2.90 

85717  :/;:! 

85893  !  ^  92 
8G243  r/f,1. 

53 
52 
51 
50 

7 
8 
9 
10 

95029 
95170 
95310 
95450 

Z.dU 

2.35 
2.34 
2.33 

95202 
9  5:  ',4  4 
95486 
95627 

2.37 
2.37 
2.36 
2.35 

53 
52 
51 
50 

11 
12 

8.86301 
86474 

2.88 
2.87 

2QA 

8.86417  fJJ 
86591  fS 

49 
48 

11 
12 

8.95589  £™ 

95728  |;^f 

8.95767 
95908 

2.34 
2.34 

49 

48 

13 
14 

15 

86645 
86816 
86987 

.oO 

2.85 
2.84 
2  83 

86763  ;•£- 
86935  fS| 
8710(i  fJJJ 

47 

45 

13 
14 
15 

960?  ^30 

%?43  :>;;;; 

96047 
96187 
96325 

2.32 
2.31 
9  Ml 

47 
46 

45 

16 

87156 

Z.oO 
209 

87277  fj! 

44 

16 

96280  1;-;^ 

96464 

230  44 

17 

87325 

•  O^j 

•'  s  1 

87447  -  •*:; 

43 

17 

D6417  -•-'_ 

96602 

Z-oU  4« 

29Q  ! 

18 

87494 

A.Ol 

•>  TO 

87616  fjf 

42 

18 

9655;;  :,--! 

96739 

*2V 

29Q 

42 

19 
20 
21 

87661 
87829 
8.87995 

a»tv 

2.78 
2.77 
2.76 

87785  fjj 
87953  f  JJj 

8.88120  f  '• 

41 

40 
39 

19 

20 
21 

9668<>  -•-;. 
9^825  -•-!; 

96877 
97013 
8.97150 

~Zv 

2.28 
2.27 

0  OA 

41 

40 
39 

22 

88161 

27  ^ 

88287  r/l_ 

38 

22 

'97095  :~*. 

97285 

*fi  .ZQ 

20  A 

38 

23 

88326 

.  to 

88453  H; 

37 

23 

9722H  ™ 

97423 

•  20 

37 

24 
25 

8S490 
88654 

2.74 
2.73 

2*7O 

88618  2-lb 
88783  fij 

36 
35 

24 
25 

97363 
9749(5 

Z.23 

8.22 

2  no 

97556 
97091 

2.25 
2.24 

2nj 

36 
35 

26 
27 

28 

88817 
88980 
89142 

.  i  £t 

2.71 

2.70 

•>  an 

88948  f'* 
89111  fi'f 
89274  fif 

34 
33 

32 

26 
27 
28 

97629 
97762 
97894 

.11 

2.21 
2.21 

t>)  on 

97825 
97959 
98092 

./4 
2.23 
2.22 
299 

34 
33 
32 

29 

89304 

z*ov 

89437  r' 

31 

29 

98026 

-l./U 

98225 

./z 

31 

30 

89464 

j  £9598  z</" 

30 

30 

98157 

2.19 

98358 

2.21 

30 

31 

8.89625 

O  (*(* 

8.89760  J-JJ 

29 

31 

8.98288 

2.18 

21  Q 

8.9841)0 

2.20 

2nn 

29 

32 

89784 

Z.OO 

89920  i  f  J" 

28 

32 

98419 

.18 

98622 

.zU 

28 

33 
34 

89943 
90102 

2.65 
2.64 

90080  J'JJ 

27 
26 

33 

34 

98549|ffJ 

98671'  J-JJ 

98753 
98884 

2.19 
2.18 

O  1Q 

27 
26 

35 

90260 

9*  '9 

90399  ^ 

25 

35 

98808 

21  ^ 

99015 

/•lo 
21  7 

25 

36 
37 

90417 
90574 

2^61 
2fiO 

j$g£8 

24 
23 

36 
37 

98937 
99066 

.10 

2.14 

214 

99145 
99275 

•  1  / 

2.16 

0  1A 

24 

23 

38 
39 

90730 
90885 

•  DU 

2.59 

2r  Q 

90872  2'62 
»ld39|fi: 

22 
21 

38 
39 

991  94 
99322 

•  i  4 

2.13 

2*1  'J 

99405  ;•;" 
99534  J-}j| 

22 
21 

40* 
41 

91040 
8.91195 

.Oo 

2.58 

91185  ';•"" 
8.91340  jrri 

20 
19 

40 
41 

99450 
8.99577 

.  J  *•> 

2.12 

99662 
8.99791 

£.J«J 

2.14 

20 
19 

42 

91349 

2.57 
9  Sfi 

91495  !;'?° 

18 

42 

99704 

2.11 
911 

m»  *i! 

18 

43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 

91502 
91655 
91807 
91959 
92110 
92261 
92411 
92561 
8.92710 
92859 
93007 

Z.OO 

2.55 
2.54 
2.53 
2.52 
2.51 
2.50 
2.49 
2.49 
2.48 
2.47 
9  4fi 

91650  ™ 
91803  f?J 

91957  frr 

92110  -"£ 
92262  f?! 
92414  J'g 
92565  f  J* 
92716  f?J 
8.92866  J-JJ 
93016  2-*9 
93165  fJJ 

17 
16 
15 
14 
13 
12 
11 
10 

g 

8 

7 

43 
44 

45 

47 

48 
49 
50 
51 
52 
53 

99830 
99956 
9.00082 
00207 
00332 
00456 
00581 
00704 
9.00828 
0095] 
01074 

^.11 
2.10 
2.09 

2.09 
2.08 
2.  (>s 
2.  (17 
2.06 
2.0P 
2.05 
2.05 

9.00046 
00174 
00301 
00427 
00553 
00679 
00805 
00930 
9.01055 
01179 
01303 

2.12 
2-12 
2.11 
2.10 
2.10 
2.09 
2.08 
2.08 
2.07 
2.07 

o  (\a 

17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 

54 
55 
56 

57 
58 
59 
60 

93154 
93301 
93448 
93594 
93740 
93885 
94030 

j2.^tO 

2.45 
2.44 
2.43 
2.43 
2.42 
2.41 

93313  2-47 

93009  ;;••[!? 
93756  £jj 
9390:;  ™ 
94049  fJJ 

94195  -  4" 

6 
5 
4 
3 

2 
1 
0 

54 
55 
56 
57 
58 
59 
60 

01196 
01318 
01440 
01561 
01682 
01803 
01923 

2^03 
2.03 
2.02 
2,02 

2.01 
2.01 

01427 
01550 
01673 
01796 
01918 
02040 
02162 

z»uo 
2.06 
2.05 
2.05 
2.04 
2.03 
2.03 

6 
5 
4 
3 
2 

1 

0 

M. 

Cosine. 

Dl"  CotaW?-  Dl" 

M. 

M. 

Cosine. 

Dl" 

Cot;ins.  PI" 

M. 

30 


SINES  AND    TANGENTS. 


M. 

Sine. 

Dl" 

Tang. 

1)1" 

M. 

M. 

Siue. 

Dl" 

Tang. 

Dl" 

M. 

0 
1 

2 

9.01923 
02043 
02163 

2.00 
2.00 

9.02162 
02283 
02404 

2.02 
2.02 

60 
59 

58 

0 
1 

2 

9.08589 
08692 
08795 

1.71 
1.71 

9.08914 
09019 
09123 

1.74 
1.73 

60 
59 

58 

3 
4 
5 

02283 
02402 
02520 

1.99 
1.98 
1.98 

02525 
02645 
02766 

2.01 
2.01 

2.00 

57 
56 
55 

3 
4 
5 

08897 
08999 
09101 

1.70 
1.70 
1.70 

09227 
09330 
09434 

.73 
1.73 
1.72 

57 
56 
55 

6 

7 

02639 

02757 

^.97 
1.97 

02885 
03005 

1.99 
1.99 

54 
53 

6 

7 

09202 
09304 

1.69 
1.69 

09537 
09640 

.72 
1.72 

54 
53 

8 
9 
10 

02874 
02992 
03109 

1.96 
1.96 
1.95 

03124 
03242 
03361 

1.98 
1.98 
1.97 

52 
51 
50 

8 
9 
10 

09405 
OU506 
09606 

1.68 
1.68 
1.68 

09742 
09845 
09947 

.  71 
1.71 
1.70 

1  ^-  A 

52 
51 
50 

11 

9.03226 

1  .95 

9.03479 

1.97 

49 

11 

9.09707 

1.67 

9.10049 

I.  IV 

49 

12 
13 

03342 
03458 

1.94 
1.94 

03597 
03714 

1.96 
1.96 

48 
47 

12 
13 

09807 
09907 

l.Ol 

1.67 

10150 
10252 

1.69 
1.69 

48 
47 

14 

03574 

1.93 

03832  i}-rr 

46 

14 

10006 

1.66 

10353 

L.69 

46 

15 
16 

03690 
03805 

1.93 
1.92 

03948  -r? 
04065  1  94 

45 
44 

15 
16 

10106 
10205 

1.66 
1.65 

10454 
10555 

1.68 
1.68 

45 
44, 

17 

03920 

1.92 

Ini 

43 

17 

10304 

1.65 

1AA 

10656 

L.68 

1A7 

43 

18 

04034 

.y  L 

1  Q  1 

04297s  r'^ 

42 

18 

10402 

.04 

1.64 

10756 

.D/ 

I  67 

42 

19 

20 

04149 
04262 

L  •  t?  L 

1.90 

04413;  * 
04528  i™ 

41 
40 

19 
20 

10501 
10599 

L64 

10856 
10956 

L67 

.41 
40 

21 

9.04376 

1.89 

9.04643  J'^f 

39 

21 

9.10697 

1.63 

9.11056 

1.66 

39 

22 

04490 

1.89 

1  88 

04758  J'JJ 

38 

22 

10795 

1.63 

ICO 

11155 

1.66 
1fi5 

38 

23 

04603 

1  .00 

04873  J'J* 

37 

23 

10893 

.Do 

11254 

.Ow 

37 

24 

04715 

1.88 

04987  !}•£ 

36 

24 

10990 

1.62 

11353 

1.65 

36 

25 

04828 

1.87 

1Q7 

05101  \'H 

35 

25 

11087 

1.62 

1A*> 

11452 

1.65 

1C  A 

35 

26 

04940 

.of 

1Q7 

05214  I'll 

34 

26 

11184 

.OZ 
1/1-1 

11551 

.0-4 

34 

27 

05052 

.87 

05828  {-3 

33 

27 

11281 

.ol 

11649 

1.64 

33 

28 
29 

05164 
05275 

K86 

05441  '88 
05553  .L8° 

32 
31 

28 
29 

11377 
11474 

1.61 
1.61 

11747 

11845 

1.64 
1.63 

32 
31 

30 

31 

05386 
J.05497 

L85 

05666  };JJ 
9.05778  |l-*i 

30 
29 

30 
31 

11570 
9.11666 

1.60 
1.60 

11943 
9.12040 

1.63 

1.62 

30 

29 

32 
33 

05607 
05717 

1.84 
1.84 

05890  |  '*' 
06002  {;JJ 

28 
27 

32 
33 

11761 

11857 

1.59 
1.59 

12138 
12235 

1.62 
1.62 

28 
27 

34 

05827 

1.83 

061131  JQ? 

26 

34 

11952 

1.55 

12332 

1.62 

26 

35 

05937 

1.83 

1QO 

06224  j1-^ 

25 

35 

1  2047 

1.58 

ICQ 

12428 

1.61 
Ir*~\ 

25 

36 

06046 

.oZ 

06335  MjT 

24 

36 

12142 

.Oc 

12525 

.ol 

24 

37 

06155 

1.82 

06445 

l.tt-4 

23' 

37 

12236 

1.58 

12621 

1.60 

23 

38 

06264 

1.81 

06556 

1.84 

22 

38 

12331 

1.57 

12717 

1.60 

22 

39 

06372 

1  .81 

06666 

1.83 

21 

39 

12425 

1.57 

12813 

1.60 

21 

40 

06481 

1.80 

06775 

1.83 

20 

40 

12519 

1.57 

12909 

1.59 

20 

41 

9.06589 

1.80 

9.06885 

1.82 

19 

41 

9.12612 

1.56 

9.13004 

1.59 

19 

42 

06696 

1.79 

06994 

1.82 

18 

42 

12706 

1.56 

13099 

1.59;  ,Q 

43 

06804 

1.79 

1  7Q 

07103 

1  w  1 

17 

43 

12799 

1.56 
1  ^ 

13194 

1.58  ,_ 

•ICQ1 

44 
45 
46 
47 

48 
49 

06911 
07018 
07124 
07231 
07337 
07442 

L.  i  VI 

1.78 
1.78 
1.77 
1.77 
1.76 

07320  !'8J 
07428  ill? 

07536  I  79 
07643  J  '™ 

07751  iH9 

16 
15 
14 
13 
12 
11 

44 
45 

46 
47 
48 
49 

12892 
12985 
13078 
13171 
13263 
13355 

l.Ot 

1.55 
1.55 
1.54 
1.54 
1.53 

13289 
13384 
13478 
13573 
13667 
13761 

1  .Do 

1.58 
1.57 
1.57 
1.57 
1.56 

16 
15 
14 
13 
12 
11 

50 

07548 

1.76 

07858 

l.i  a 

10 

50 

13447 

1.5.' 

1  3854 

1.56 

10 

51 

9.07653 

1.75 

9.07964 

l.7£ 

9 

51 

9.13539 

1.51 

9.13948 

1.56 

9 

52 

07758 

1.75 

08071  \'ll 

8 

52 

13630 

1.52 

14041 

1.55 

8 

53 

07863 

1.75 

08177 

I'ml  i 

7 

53 

13722 

1.52 

14134 

1.55 

7 

54 

07968 

1.74 

OS283 

1.71 

6 

54 

13813 

1.52 

14227 

1.55 

6 

55 

08072 

1.74 

08389 

1.7( 

5 

55 

1  3904 

1.52 

14320 

1.55 

5 

56 

08176 

1.72 

08495 

1.7( 

4 

56 

13994 

1.51 

14412 

1.54 

4 

57 

08280 

1  .72 

08600 

1.75 

3 

57 

14085 

1.51 

14504 

1.54 

3 

58 

08383 

1.72 

08705 

1.75 

2 

58 

14175 

1.51 

14597 

1.54 

2 

59 

08486 

1.75 

08810 

1.75 

1 

59 

14266 

1.56 

14688 

1.53 

1 

60 

08589 

1.72 

08914 

1.74 

0 

60 

14356 

1.50 

14780 

1.53 

0 

IT 

Cosine. 

Dl" 

Cotang. 

Dl" 

M. 

M. 

CnsilK'. 

Dl" 

Cotang. 

Dl" 

M. 

31 


82° 


8° 


TABLE  IV.— LOGARITHMIC 


9 


M. 

Sine. 

1)1'' 

Tang. 

1)1" 

M. 

31. 

Sine. 

Dl" 

Tang. 

Dl" 

31. 

0 

9.14356 

9.14780 

60 

0 

9.19433 

9.19971 

60 

1 

14445 

1.50 

14872 

1.53 

59 

19513 

.33 

20053 

1.36 

59 

2 

14535 

1.49 

14963 

1.52 

58 

2 

19592 

.33 

20134 

1.36 

58 

3 

14624 

1.49 

15054 

1.52 

57 

3 

19672 

.32 

20216 

1.36 

57 

4 
5 

14714 
14803 

1.49 
1.48 

15145 
15236 

1.52 
1.51 

56 
55 

4 
5 

19751 
19830 

.32 
.32 

20297 
20378 

1.35 
1.35 

56 
55 

6 

7 

14891 
14980 

1.48 
1.48 

15327 
15417 

1.51 
1.51 

54 
53 

6 

7 

19909 

19988 

.32 
.31 

20459 
20540 

1.35 
1.35 

54 
53 

8 

15069 

1.48 

15508 

1.50 

52 

8 

20067 

.31 

20621 

1.35 

52 

9 

15157 

1.47 

15598 

1.50 

51 

9 

20145 

.31 

20701 

1.34 

51 

10 

15245 

1.47 

15688 

.50 

50 

10 

20223 

.31 

20782 

1.34 

50 

11 

9.15333 

1.47 

9.15777 

.50 

49 

11 

9.20302 

.30 

9.20862 

1.34 

49 

12 

15421 

1.46 

15867 

.49 

48 

12 

20380 

.30 

20942 

1.33 

48 

13 

15508 

1.46 

15956 

.49 

47 

13 

20458 

.30 

21022 

1.33 

47 

14 
15 

15596 
15683 

1.46 
1.45 

16046 
16135 

.49 

.48 

46 
45 

14 
15 

20535 
20613 

.30 
.29 

21102 
21182 

1.33 
1.33 

46 
45 

16 

15770 

1.45 

14  ti 

16224 

.48 

A  U 

44 

16 

20691 

.29 
oo 

21261 

1.33 

1QO 

44 

17 

15857 

.40 

16312 

.4o 

43 

17 

20768 

.zy 

21341 

,oZ 

43 

18 

15944 

1.45 

16401 

.48 

1  7 

42 

18 

20845 

.29 

oo 

21420 

1.32 

Too 

42 

19 
20 

16030 
16116 

1.44 
1.44 

16489 
16577 

,4/ 

.47 

41 

40 

19 

20 

20922 
20999 

•  Zo 

.28 

21499 
21578 

.0.6 

1.32 

41 
40 

21 

9.16203 

1.44 

9.16665 

.47 

39 

21 

9.21076 

.28 

9.21657 

1.32 

39 

22 

16289 

1.43 

16753 

.46 

38 

22 

21153 

.28 

21736 

1.31 

38 

23 

16374 

1.43 

16841 

.46 

37 

23 

21229 

.27 

21814 

1.31 

37 

24 
25 
26 

16460 
16545 
16631 

1.43 
1.42 
1.42 

16928 
17016 
17103 

.46 
.46 
.45 

36 
35 
34 

24 
25 

26 

21306 
21382 
21458 

.27 
.27 
.27 

21893 
21971 
22049 

1.31 
1.31 
1  .30 

36 
35 

34 

27 

16716 

1.42 

17190 

.45 

33 

27 

21534 

.27 

OC 

22127 

1.30 

33 

28 
29 

16801 
16886 

1.42 
.41 

17277 
17363 

.45 
.44 

32 
31 

28 
29 

21610 
21685 

.ZO 

.26 

22205 

22283 

1  .30 
1.30 

32 
31 

30 

16970 

1.41 

17450 

.44 

30 

30 

21761 

.26 

22361 

1.29 

30 

31 

9.17055 

1.41 

9.17536 

.44 

29 

31 

9.21836 

.26 

9.22438 

1  .29 

29 

32 
33 
34 
35 
36 

17139 
17223 
17307 
17391 
17474 

11.40 
1.40 
1.40 
1.40 
1.39 

17622 
17708 
17794 

17880 
17965 

.44 
.43 
.43 
.43 
.42 

28 
27 
26 
25 
24 

32 
33 
34 
35 
36 

21912 

21987 
22062 
22137 
22211 

!25 
.25 
.25 

22516 
22593 
22670 
22747 

22824 

1.29 
1.29 
1.29 
1.28 
1.28 

28 
27 
26 
25 
24 

37 

38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 

17558 
17641 
17724 
17807 
9.17890 
17973 
18055 
18137 
18220 
18302 
18383 
18465 
18547 
18628 
9.18709 
18790 
18871 
18952 

1.39 
1.39 
1.39 
1.38 
1.38 
1.38 
1.37 
1.37 
1.37 
1.37 
1.36 
1.36 
1.36 
1.36 
1.35 
1.35 
1.35 
1.35 

18051 
18136 
18221 
18306 
9.18391 
18475 
18560 
18644 
18728 
18812 
18896 
18979 
19063 
19146 
9.19229 
19312 
19395 
19478 

.42 
.42 
.42 
.42 
1.41 
1.41 
1.41 
1.40 
.40 
1.40 
1.40 
1.39 
.39 
1.39 
1.39 
1.38 
1.38 
1.38 

23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 

37 

38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 

22286 
22361 
22435 
22509 
9.22583 
22657 
22731 
22805 
22878 
22952 
23025 
23098 
23171 
23244 
9.23317 
23390 
234(52 
23535 

.24 
.24 
.24 
.24 
.24 
.23 
1.23 
1.23 
1.23 
1.22 
1.22 
1.22 
1.22 
1.22 
1.21 
1.21 
1.21 
1.21 

22901 
22977 
23054 
23130 
9.23206 
23283 
23359 
23435 
23510 
23586 
23661 
23737 
23812 
23887 
9.23962 
24037 
24112 
24186 

1.28 
1.28 
1.28 
1.27 
1.27 
1.27 
1.27 
1.27 
1.26 
1.26 
1.26 
1.26 
1.25 
1.25 
1.25 
1.25 
1.25 
1.24 

23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 

55 

19033 

1.34 

19561 

1.38 

IO«T 

5 

55 

23607 

.20 
9ft 

24261 

1.24 

194 

5 

56 
57 
58 
59 
60 

19113 
19193 
19273 
19353 
19433 

1.34 
1.34 
1.34 
1.33 
1.33 

19643 
19725 
19807 
19889 
19971 

•Of 

1.37 
1.37 

1.37 
1.36 

4 
3 
2 
1 
0 

56 
57 
58 
59 
60 

23679 
23752 
23823 
23895 
23967 

.zu 
.20 
.20 
.20 
.20 

24335 
24410 
24484 
24558 
24632 

./* 

1.24 
1.24 
1.23 
1.23 

4 
3 
2 
1 
0 

M. 

Cosine. 

Dl" 

Cotang. 

Dl" 

M. 

M. 

Cosine. 

Dl" 

Cotang. 

Dl" 

M. 

81C 


32 


SINES  AND  TANGENTS. 


11 


M. 

Sine. 

Dl" 

Tiing. 

Dl" 

M. 

M. 

Sine. 

Dl" 

Tang. 

Dl" 

M. 

0 

9.23967 

11  0 

9.24632 

1.)  .. 

60 

0 

9.28060 

9.28865 

11  O 

60 

1 

24039 

.  i  y 

24706 

.20 

59 

1 

28125 

1.08 

28933 

.12 

59 

2 

24110 

1.19 

24779 

1.23 

58 

2 

28190 

1.08 

29000 

1.12 

58 

3 

24181 

1.19 

11  u 

24853 

1.23 
100 

57 

3 

28254 

1.08 

29067 

1.12 

1  o 

57 

4 

24253 

.  i  y 

24926 

.22 

56 

4 

28319 

1.08 

29134 

.1  z 

56 

5 
6 

7 

24324 
24395 
24466 

1.18 
1.18 
1.18 

25000 
25073 
25146 

1.22 
1.22 
1.22 

55 
54 
53 

5 
6 

7 

28384  i 
28448  1 
28512 

1.08 
1.07 
1.07 

29201 
29268 
29335 

.12 
.12 
.11 

55 
54 
53 

8 

24536 

1.18 

25219 

1.22 

52 

8 

28577 

1.07 

29402 

.11 

52 

9 

24607 

1.18 

25292 

1.22 

51 

9 

28641 

1.07 

29468 

.11 

51 

10 

24677 

1.17 

25365 

1.21 

50 

10 

28705 

1.07 

29535 

.11 

50 

11 

9.24748 

1.17 

9.25437 

1.21 

49 

11 

9.28769 

1.07 

9.29601 

.11 

49 

12 

24818 

1.17 

25510 

1.21 

48 

12 

28833 

1.06 

29668 

.11 

48 

13 
14 
15 
16 

24888 
24958 
25028 
25098 

1.17 
1.17 
1.17 
1.16 

25582 
25655 
25727 
25799 

1.21 
1.20 
1.20 
1.20 

47 
46 
45 
44 

13 
14 
15 

16 

28896 
28960 
29024 
29087 

1.06 
1.06 
1.06 
.06 

29734 
29800 
29866 
29932 

.10 
.10 
.10 
.10 

47 
46 
45 
44 

17 

25168 

1.16 

25871 

1.20 

43 

17 

29150 

.06 

29998 

.10 

43 

18 

25237 

1.16 

25943 

1.20 

42 

18 

29214 

.05 

30064 

.10 

42 

19 

25307 

1.16 

26015 

1.20 

41 

19 

29277 

.05 

30130 

.10 

41 

20 

25376 

1.16 

26086 

1.19 

40 

20 

29340 

.05 

30195 

.09 

40 

21 

22 

9.25445 
25514 

1.15 
1.15 

9.26158 

26229 

1.19 
1.19 

39 

38 

21 
22 

9.29403 
29466 

.05 
.05 

9.30261 
30326 

.09 
.09 

39 
38 

23 

25583 

1.15 

26301 

1.19 

37 

23 

29529 

.05 

30391 

.09 

37 

24 
25 

25652 
25721 

1.15 
1.15 

26372 
26443 

1.19 
1.18 

36 
35 

24 

25 

29591 
29654 

.04 
.04 

30457 
30522 

.09 
.09 

36 
35 

26 

25790 

1.14 

26514 

1.18 

34 

26 

29716 

.04 

30587 

.08 

34 

27 

25858 

1.14 
11  \ 

26585 

1.18 

1  1  Q 

33 

27 

29779 

.04 

30652 

.08 

0,8 

33 

28 

25927 

.  IT: 

26655 

1  .  lo 

32 

28 

29841 

.U4 

30717 

.Uo 

32 

29 
30 

25995 
26063 

1.14 
1.14 

26726 
26797 

1.18 
1.18 

31 
30 

29 

30 

29903 
29966 

.04 
.04 

30782 
30846 

.08 
.08 

31 
30 

31 

9.26131 

1.13 

9.26867 

1.17 

29 

31 

9.30028 

.03 

9.30911 

.08 

29 

32 

26199 

1.13 

110 

26937 

1.17 

117 

28 

32 

30090 

.03 

AO 

30975 

.07 

28 

33 

26267 

1  .  IO 
11  O 

27008 

1  .  1  1 

27 

33 

30151 

.Uo 

31040 

.07 

27 

34 

26335 

.id 

27078 

1.17 

26 

34 

30213 

.03 

31104 

.07 

26 

35 

26403 

1.13 

27148 

1.17 

25 

35 

30275 

.03 

31168 

.07 

25 

36 

26470 

1.13 

27218 

1.17 

11  G 

24 

36 

30336 

.03 

31233 

.07 

24 

37 

38 

26538 
26605 

1.12 
1.12 

27288 
27357 

.16 
1.16 

23 
22 

37 

38 

30398 
30459 

.03 
.02 

31297 
31361 

.07 
.07 

23 

22 

39 
40 

26672 
26739 

1.12 
1.12 

27427 
27496 

1.16 
1.16 

21 
20 

39 
40 

30521 
30582 

.02 
.02 

31425 
31489 

.07 
.06 

21 
20 

41 
42 

9.26806 
26873 

1.12 
1.12 

9.27566 
27635 

1  .16 
1.15 

19 

18 

41 

42 

9.30643 
30704 

.02 
.02 

9.31552 
31616 

.06 
.06 

19 
18 

43 

26940 

1.11 

27704 

1.15 

17 

43 

30765 

.02 

31679 

.06 

17 

44 

V27007 

1.1  1 
ill 

27773 

1.15 

11  E 

16 

44 

30826 

.01 

Al 

31743 

.06 

16 

45 

27073 

1.11 

27842 

.10 

15 

45 

30887 

.U  1 

31806 

.06 

15 

46 

27140 

1.11 
1-i  t 

27911 

1.15 

14 

46 

30947 

.01 

A  1 

31870 

.05 

A  £ 

14 

47 

27206 

.1  L 

11  (\ 

27980 

1.15 
11  ^ 

13 

47 

31008 

.U  1 

Al 

31933 

,uo 

A  & 

13 

48 
49 

27273 
27339 

.1  U 

1.10 

28049 
28117 

.10 

1.14 

12 
11 

48 
49 

31068 
31129 

.Ul 

.01 

31996 
32059 

.UO 

.05 

12 
11 

50 

27405 

1.10 

28186 

1.14 

10 

50 

31189 

.01 

32122 

.05 

10 

51 

9.27471 

1.10 

11  {\ 

9.28254 

1.14 

9 

51 

9.31250 

.00 

AA 

9.32185 

1.05 
j\f 

9 

52 

27537 

.1U 

28323 

1.14 

8 

52 

31310 

.uu 

32248 

.UO 

8 

53 
54 

55 
56 

57 

27602 
27668 
27734 
27799 

27864 

1.10 
1.09 
1.09 
1.09 
1.09 

28391 
28459 

28527 
28595 
28662 

1.14 
1.13 
1.13 
1.13 
1.13 

7 
6 
5 
4 
3 

53 
54 
55 
56 

57 

31370 
31430 
31490 
31549 
31609 

.00 
.00 
.00 
.00 
1.00 

32311 
32373 
32436 
32498 
32561 

.05 
.04 
.04 

.04 
.04 

7 
6 
5 
4 
3 

58 
59 

27930 
27995 

1.09 
1.09 

28730 
28798 

1.13 
1.13 

2 
1 

58 
59 

31669 
31728 

.99 
.99 

32623 
32685 

.04 
1.04 

2 
1 

60 

28060 

1.08 

28865 

1.12 

0 

60 

31788 

.99 

32747 

1.04 

0 

M. 

Cosine. 

Dl" 

Cotangr. 

Dl" 

M. 

M. 

Cosine. 

Dl" 

Cotang. 

Dl" 

M. 

79° 


78C 


TABLE  IV.— LOGARITHMIC 


M. 

Sine. 

Di" 

Tang. 

1)1" 

M. 

M. 

Sine. 

1)1" 

Taug. 

Dl" 

M. 

0 
1 

9.31788 
31847 

0.99 

9.32747 
32810 

1.03 

60 
59 

0 
1 

9.35209 
35263 

0.91 

9.36336 
36394 

0.96 

60 
59 

2 

31907 

.99 

on 

32872 

1.03 
In  •! 

58 

2 

35318 

.91 
ni 

36452 

.96 

58 

3 

31966 

.yy 

32933 

jVo 

57 

3 

35373 

.y  i 

3C.509 

.96 

57 

4 
5 

32025 
32084 

.99 

.98 

32995 
33057 

1.03 
1.03 

56 
55 

4 
5 

35427 
35481 

.91 
.91 

36566 
36624 

.96 
.96 

56 
55 

6 

7 

32143 
32202 

.98 
.98 

33119 

33180 

1.03 
1.03 

1AO 

54 
53 

6 

7 

35536 
35590 

.91 

.90 

36681 
36738 

.95 
.95 

f\c 

54 
53 

8 
9 
10 
11 
12 
13 

32261 
32319 
32378 
9.32437 
32495 
32553 

!98 
.98 
.98 
.97 
.97 

33242 
33303 
33365 
9.33426 
33487 
33548 

.06 

1.02 
1.02 
1.02 
1.02 
1.02 

52 
51 
50 
49 
48 
47 

8 
9 
10 
11 
12 
13 

35644 
35698 
35752 
9.35806 
35860 
35914 

!QO 

.90 
.90 
.90 
.90 

36795 
36852 
36909 
9.36966 
37023 
37080 

.yo 
.95 
.95 
.95 
.95 
.95 

52 
51 
50 
49 
48 
47 

14 
15 
16 

32612 
32670 
32728 

.97 
.97 

.97 

33609 
33670 
33731 

1.02 
1.02 
1.01 

46 
45 
44 

14 
15 
16 

35968 
36022 
36075 

.90 
.89 
.89 

37137 
37250 

.95 
.94 

.'.14 

46 
45 
44 

17 
18 
19 

32786 
32844 
32902 

.97 
.97 
.97 

Qfi 

33792 
33853 
33913 

1.01 
1.01 
1.01 

1A  1 

43 
42 
41 

17 

18 
19 

36129 
36182 
36236 

•89 
•89 
.89 

CO 

37306 
37363 
37419 

.94 
.94 

43 
42 
41 

20 
21 

32960 
9.33018 

«yo 
.96 

33974 
9.34034 

.01 
1.01 

40 
39 

20 
21 

36289 
9.36342 

•by 
.89 

37476 
9.37532 

.94 

40 

39 

22 

33075 

34095 

1.01 

AI 

.38 

22 

36395 

on 

37588 

.94 

n  i 

38 

23 

33133 

Qii 

34155 

.0  1 

37 

23 

36449 

•by 

37644 

.,14 
no 

37 

24 

33190 

.yo 

34215 

.00 

nn 

36 

24 

36502 

•88 

QO 

37701) 

.y,> 

36 

25 

33248 

OA 

34276 

.00 

35 

25 

36555 

•OO 

37756 

.y.i 

35 

26 

33305 

.yo 

n  \ 

34336 

.00 

An 

34 

26 

36608 

•88 

QO 

37812 

.93 

34 

27 

33362 

.y  o 

34396 

.Uu 

33 

27 

36660 

•OO 

37868 

.93 

33 

28 

33420 

_ 

34456 

.00 

32 

28 

36713 

•88 

37924 

.93 

32 

29 

33477 

•  9o 

34516 

.00 
no 

31 

29 

36766 

•  88 

QO 

37980 

.93 

31 

30 

33534 

.95 

34576 

.UU 

on 

30 

30 

36819 

•OO 
QQ 

38035 

os 

30 

31 

9.33591 

9.34635 

•  UU 

29 

31 

9.36871 

•oo 

9.38091 

•*  * 

29 

32 

33647 

•95 

34695 

.99 

28 

32 

36924 

•88 

87 

38147 

.93 

28 

33 

33704 

34755 

.99 

27 

33 

36976 

•O  4 

38202 

•  JZ 

27 

34 

33761 

n  i 

34814 

.99 

26 

34 

37028 

•87 

38257 

.92 

26 

35 

33818 

•  94 
•  94 

34874 

.99 
qq 

25 

35 

37081 

•87 

38313 

.92 

25 

36 

33874 

34933 

.yy 

24 

36 

37133 

•o< 

38368 

•"* 

24 

37" 

33931 

.94 

34992 

.99 

23 

37 

37185 

•87 

38423 

'  .92 

23 

38 

33987 

•94 

35051 

".99 
qq 

22 

38 

37237 

•87 
07 

38479 

'.92 

22 

39 

34043 

35111 

.yy 

21 

39 

37289 

•O  i 

38534 

•V* 

21 

40 

34100 

•94 

35170 

.98 

20 

40 

37341 

•87 

38589 

.92 

20 

41 

9.34156 

•94 

9.35229 

.98 

19 

41 

9.37393 

•87 

9.38644 

.92 

19 

42 

34212 

**' 

35288 

.98 

no 

18 

42 

37445 

•86 

Qrt 

38699 

.92 

18 

43 

34268 

'*:* 

35347 

.9o 

17 

43 

37497 

•  OO 

38754 

.yi 

17 

44 

34324 

!> 

35405 

.98 

16 

44 

37549 

.86 

38808 

.91 

16 

45 

34380 

•93 
•93 

35464 

.98 

15 

45 

37600 

.86 

QC 

38863 

.91 
Q1 

15 

46 

34436 

35523 

•yo 

14 

46 

37652 

.00 

38918 

•  •'  1 

14 

47 

34491 

•  93 
•  93 

35581 

.98 

13 

47 

37703 

.86 

Qi! 

38972 

.91 

Q1 

13 

48 

34547 

35640 

.9o 

12 

48 

37755 

•  oO 

Q/> 

39027 

.y  i 

fit 

12 

49 

34602 

•  J6 

35698 

.97 

11 

49 

37806 

.OO 

39082 

.yi 

11 

50 

34658 

no 

3575.7 

.97 

10 

50 

37858 

.86 

39136 

.91 

10 

51 

9.34713 

•  31 
qo 

9.35815 

.97 

n*7 

9 

51 

9.37909 

.85 

QC 

9.39190 

.91 
01 

9 

52 

34769 

•  y  Zi 

35873 

•  y  t 

8 

52 

37960 

•  OO 

39245 

.y  i 

8 

53 

34824 

•92 

35931 

.97 

7 

53 

38011 

.85 

39299 

.90 

7 

54 

34879 

•yz 

35989 

.97 

6 

54 

38062 

.85 

39353 

.90 

6 

55 

34934 

•92 

no 

36047 

.97 

5 

55 

381  1  3 

•85 

39407 

.90 

5 

56 

34989 

•  92 

36105 

.97 

4 

56 

38164 

.85 

39461 

.90 

4 

57 

35044 

•92 

36163 

.96 

3 

57 

38215 

.85 

39515 

.90 

3 

58 
59 

35099 
35154 

•92 
•91 

36221 
36279 

.96 
.96 

2 
1 

58 
59 

38266 
38317 

.85 
.85 

39569 
39623 

.90 
.90 

2 
1 

60 

35209 

.9] 

36336 

.96 

0 

60 

38368 

.85 

39677 

.90 

0 

M. 

Cosine. 

Dl" 

Cotang. 

Dl" 

M. 

M. 

Cosine. 

Dl" 

CutitllU. 

Dl" 

M. 

77; 


14C 


SINES  AND    TANGENTS. 


I5C 


M. 

Sine.   Dl" 

Tan-. 

1)1"  |  M. 

M. 

Sine. 

Dl" 

Tang. 

1)1" 

M. 

0 
1 

9.38368 
38418 

0.84 

y.:',w>77 
39731 

0.90 
un 

M 

59 

1 

9.41300 
41347 

0.79 

.78 

9.42805 

42S50 

0.84 
.84 

60 

59 

2 

38469 

.84 

39785 

.yu 

Qf\ 

58 

2 

4  1  394 

42906 

58 

3 

38519 

.84 

39838 

.89 

f>7 

;; 

41441 

.78 

42957 

.84 

57 

4 
5 
fi 

38570 
38620 

38670 

.84 
.84 
.84 

39892 
39945 
39999 

!89 

.89 

50 
5  5 
54 

4 

5 
0 

41488 
41535 
41582 

.78 
.78 
.78 

43007 
43057 
43108 

M  »« 

[ 

7 

3S721 

.84 

Q  1 

40052 

.89 

0(1 

53 

7 

41628 

.78 
170 

43158 

.84 
84 

53 

8 
9 

38771 
38821 

.o4 

.84 

QQ 

40106 
40159 

.  o  J 

.89 

52 
51 

8 
9 

41675 
41722 

.  t  o 

.78 
7ft 

43208 
43258 

.O^r 

.83 

QO 

52 
51 

10 

38871 

.OO 
00 

40212 

on 

50 

10 

41768 

.  1  O 

70 

43308 

.OO 
QO 

50 

11 
12 

9.38921 

38971 

.OO 

.83 

9.40266 
40319 

.oy 
.89 

49 

48 

11 
12 

9.41815 

4  1  SO  1 

•  I  O 

.78 

9.43358 
43408 

.OO 

.83 

49 

48 

13 
14 

39021 
39071 

.83 
.83 

80 

40372 
40425 

.88 
.88 

00 

47 
46 

13 

14 

41908 
41954 

.77 

.77 
77 

43458 

43508 

.83 
.83 

QO 

47 
46 

15 

39121 

0 
QQ 

40478 

.So 

QQ 

45 

15 

42001 

•  •I 

43558 

•  OO 
QO 

45 

10 
17 

39170 
39220 

.  OO 

.83 

QO 

40531 

40584 

.  OO 

.88 

QQ 

44 
43 

16 
17 

42047 
42093 

[77 

43607 
43657 

•  oo 

.83 

83 

44 
43 

18 
19 

39270 
39319 

.80 

.83 

89 

40036 
40689 

.  OO 

.88 

QQ 

42 
41 

18 
19 

42140 
42186 

!77 

77 

43707 
43756 

!83 

QO 

42 
41 

20 
21 

39369 

9.39418 

& 

.82 

40742 
9.40795 

.  oo 

.88 

40 
39 

20 
21 

42232 
9.42278 

•  i  § 

.77 

.  43806 
9.43855 

.OO 

.83 

40 
39 

22 

39467 

.82 

40847 

.88 

QQ 

38 

22 

42324 

.77 

43905 

89 

38 

23 

39517 

•o« 

40900 

.OO 

37 

23 

42370 

.  i  ( 

43954 

,oZ 

37 

24 

39566 

.82 

no 

40952 

.87 

8*7 

36 

24 

42416 

.76 

7fi 

44004 

.82 

36 

25 

39615 

.04 

41005 

1 

35 

25- 

•  42461 

.  i  0 

44053 

.bZ 

35 

26 

27 

39664 
39713 

.82 
.82 

41057 
41109 

.87 
.87 

34 
33 

26  ' 

27 

42507 
42553 

.76 
.76 

44102 
44151 

.82 
.82 

34 
33 

28 

39762 

.82 

41161 

•  87 

32 

28 

42599 

.76 

>-/» 

44201 

.82 

32 

29 

39811 

.82 

41214 

.87 

31 

29 

42644 

.  1  O 

44250 

.82 

31 

30 

39860 

.81 

41266 

.87 

30 

30 

42690 

.76 

44299 

.82 

30 

31 

9.39909 

.81 

9.41318 

.87 

29 

31 

9.42735 

'!?i  9.44348 

.82 

29 

32 

39958 

.81 

41370 

.87 

28 

32 

42781 

•  7o 

44397 

.82 

28 

33 

40006 

.81 

41422 

.87 

27 

33 

42826 

.76 

44446 

.82 

27 

34 

40055 

.81 

41474 

.87 

26 

34 

42872 

.76 

44495 

.81 

26 

35 

40103 

.81 

41526 

.86 

25 

35 

42917 

.76 

44544 

.81 

25 

36 

40152 

.81 

41578 

.86 

24 

36 

42962 

.75 

44592 

.81 

24 

37 

40200 

.81 

41629 

.86 

23 

37 

43008 

.75 

44641 

.81 

23 

38 

40249 

.81 

41681 

.86 

22 

38 

43053 

.75 

44690 

•  81 

22 

39 

40297 

.81 

41733 

.86 

21 

39 

43098 

.75 

44738 

.81 

21 

40 

40346 

.81 

41784 

.86 

20 

40 

43143 

.75 

44787 

.81 

20 

41 
42 
43 

9.40394 
40442 

40491) 

.80 
.80 
.80 

9.41836 

41887 
41939 

.86 
.86 
.86 

19 
18 
17 

41 

42 
43 

9.43188 
43233 
43278 

.75 

.75 
.75 

9.44836 

44884 
44933 

.81 
.81 

.81 

19 
18 
17 

44 

40538 

.80 

41990 

.86 

16 

44 

43323 

.75 

44981 

.81 

16 

45 

40586 

.80 

Q  A 

42041 

.86 

15 

45 

43367 

.75 

45029 

.81 

Q  1 

15 

46 

40634 

.Oil 

42093 

.85 

14 

46 

43412 

.75 

45078 

•  ol 

14 

47 

48 

40682 
40730 

.80 
.80 

42144 
42195 

.85 
.85 

13 
12 

47 

48 

43457 
43502 

.75 
.74 

45126 
45174 

.80 
.80 

13 
12 

49 

40778  i  '*!! 

42246 

.85 

11 

49 

43546 

.74 

45222 

.80 

11 

50 

40825 

.»(» 

42297 

.85 

10 

50 

43591 

.74 

45271 

.80 

10 

51 

9.40873 

.79 

9.42348 

.85 

9 

51 

9.43635 

.74 

9.45319 

.80 

9 

52 

40921 

.79 

42399 

.85 

8 

52 

43680 

.74 

45367 

.80 

8 

53 

40968 

.79 

42450 

.85 

7 

53 

43724 

.74 

45415 

.80 

7 

54 

41016 

.79 

42501 

.85 

6 

54 

43709 

.74 

45463 

.80 

6 

55 

41063 

.79 

42552 

.85 

5 

55 

43813 

.74 

45511 

.80 

5 

56 

41111 

.79 

42603 

.85 

4. 

56 

43857 

.74 

45559 

.80 

4 

57 

41158 

.79 

42653 

.85 

3 

57 

43901 

.74 

45606 

.80 

3 

58 

41205 

.79 

42704 

.84 

2 

58 

43946 

.74 

45654 

.80 

2 

59 

41252 

.79 

42755 

.84 

1 

59 

43990 

.74 

45702 

.80 

1 

60 

41300 

.79 

42805  ' 

0 

60 

44034 

.73 

45750 

.80 

0 

M. 

Cosine. 

Dl"  Cotansr.  Dl" 

M. 

M.   Cosine.   IM" 

Cotansr. 

DP 

M. 

74C 


16C 


TABLE  IV.— LOGARITHMIC 


17° 


M.    Sine.   Dl"   Tuna.   Dl"   .M. 

M. 

Sine. 

Dl"   Tmitr.   Dl" 

11. 

0 
1 

9.44034 
44078 

0.73 
70 

9.45750 
45797 

1 

60 
59 

0 

1 

9.46594 
46635 

0.69 

|    /><» 

il.  48534 
48579 

0.75 

60 
59 

2 

44122 

.  id 
70 

45845 

70 

58 

2 

46676 

.0,' 
ft  11 

48624 

.7  ) 

58 

3 

44160 

.  I  <l 

7'J 

45892 

• 

70 

57 

3 

45717 

.0  *J 
rfq 

48669 

.75 

57 

4 

44210 

.IO 

45940 

•  •  • 

T  A 

56 

4 

46758 

•  OV4 

48714 

.7 

56 

5 

44253 

70 

45987  '11 

55 

5 

46800 

fiO 

48759 

.7 

55 

6 

44297 

•  1  »> 

7°. 

46035 

7Q 

54 

6 

46841 

•  Do 

CO 

48804  'I 

54 

7 

44341 

•  /  «  > 

46082 

»•  *• 

53 

7 

46882 

.Do 

48849 

.<0 

53 

8 
9 

44385 
4442* 

.73 
.73 

46130 
46177 

.79 
.79 

52 
51 

8 
9 

46923 
46964 

.68 

.6* 

48894 
48939 

.75 
.75 

52 
51 

10 

44472 

.73 

tjn 

46224 

.79 

TO 

50 

10 

47005 

.68 

48984 

.75 

50 

11 

9.44516 

""  9.46271 

.  i  y 

49 

11 

9.47045 

.68 

9.49029 

.75 

49 

12 

44559 

.7.1 
79 

46319 

.79 

48 

12" 

47086 

.68 

49073 

.75 

48 

13 

44602 

46366 

.<y 

47 

13 

47127 

.68 

49118 

.74 

47 

14 

44646 

.72 

46413 

.78 

46 

14 

47168  •'* 

49163 

.74 

46 

15 

44689 

.72 

TO 

46460 

.78 

45 

15 

47209 

49207 

.74 

45 

16 

44733 

.12, 

46507 

.78 

44 

16 

47249  i  '™ 

49252 

.74 

44 

17 

44776 

.72 

46554 

.78 

43 

17 

47290 

.GO 

49296 

.74 

43 

18 

44819 

.72 

46601 

.78 

TO 

42 

18 

47330 

.68 

49341 

.74 

42 

19 

44862 

.72 

46648 

.7o 

41 

19 

47371 

.68 

49385 

.74 

41 

20 

44905 

.72 

46694 

.78 

40 

20 

47411 

.67 

49430 

.74 

40 

21 

22 

9.441)48 
44992 

.72 
.72 

9.4(5741 

46788 

.78 
.78 

39 
38 

21 
22 

9.47452  '[?i  9.49474 
47492i  '"ll  49519 

.74 
.74 

39 
38 

23 

45035 

.72 

46835 

.78 

rrO 

37 

23 

47533 

•Ji   49563 

.74 

37 

24 

45077 

.72 

46881 

.78 

36. 

24 

47573 

•  1   49607 

.74 

36 

25 

45120 

.71 

46928 

.78 

35 

25 

47613 

•J-   49652 

.74 

35 

26 

45163 

.71 

46975 

.78 

34 

26 

47654 

•J;  4U696 

.74 

34 

27 

45206 

.71 

47021 

.78 

33 

27 

47694 

.67 

49740 

.74 

33 

28 

45249 

.71 

47068 

.77 

32 

28 

47734:  'JJ 

49784 

.74 

32 

29 

45292 

.71 

47114 

.77 

31 

29 

47774  '"• 

49828 

.73 

31 

30 

45334 

.71 

47160 

.77 

30 

30 

47814 

.07 

49872 

.73 

30 

31 

9.45377 

.71 

9.47207 

.77 

29 

31 

9.47854 

.67 

9.49916 

.73 

29 

32 

45419 

.71 

47253 

.77 

28 

32 

47894 

•5?l  49960 

.73 

28 

33 
34 

45462 
45504 

.71 
.71 

47299 
47346 

.77 
.77 

27 
26 

33 
34 

47934  -JJ   50004 
479741  •5il  50048 

.73 
.73 

27 
26 

35 

45547 

.71 

47392 

.77 

25 

35 

48014 

•JJ   50092 

.73 

25 

36 

45589 

.71 

47438 

.77 

24 

36 

48054 

•JJ   50136 

.73 

24 

37 

45632 

.71 

47484 

.77 

23 

37 

48094 

•JJ  50180 

.73 

23 

38 

45674 

.71 

47530 

.77 

22 

38 

48133 

•JJ   50223 

.73 

22 

39 

45716 

.70 

47576 

.77 

21 

39 

48173 

•JJ   50267 

.73  21 

40 

45758 

.70 

47622 

.77 

20 

40 

48213 

•(!b   50311 

•JJ  20 

41 

9.45801 

.70 

9.47668 

.77 

19 

41 

9.48252 

•JJi  9.50355 

.73  19 

42 

45843 

.70 

47714 

.77 

18 

42 

48292 

.66 

50398 

.73  ]8 

43 

45885 

.70 

47760 

.76 

17 

43 

48332 

.66 

50442 

.73  ,. 

44 

45927 

.70 

47806 

.76 

16 

44 

48371 

.66 

50485 

.73  16 

45 

45969 

.70 

47852 

.76 

15 

45 

48411 

.66 

50529 

"°  15 

46 

46011 

.70 

47897 

.76 

14 

46 

48450 

.66 

50572 

•11  14 

47 

46053 

.70 

47943 

.76 

13 

47 

48490 

.66 

50616 

•  '  -  13 

48 

46095 

.70 

47989 

.76 

12 

48 

48529 

.66 

50659 

'~-l  12 

49 

46136 

.70 

48035 

.76 

11 

49 

48568 

.66 

50703 

50 
51 

46178 
9.46220 

.70 
•11 

48080 
9.48126 

.76 
.76 

10 

9 

50 
51 

48607 
9.48647 

•JJ   50746 
•JJ  9.50789 

'-2  10 

52 

46262 

.69 

48171 

.76 

8 

52 

48686 

•JJ   50833 

''11   8 

53 

46303 

.69 

48217 

.76 

7 

53 

48725 

.65 

50876 

.72 

7 

54 

46345 

.69 

48262 

.76 

6 

54 

4S764 

.65 

50919 

.72 

6 

55 

46386 

.69 

48307 

.76 

5 

55 

48803 

.65 

50962 

.72 

5 

56 

46428 

.69 

48353 

.76 

4 

.56 

48812 

.65 

51005 

.72 

4 

57 

46469 

.69 

48398 

.76 

3 

57 

48881 

.65 

51048 

.72 

3 

58 

46511 

.69 

48443 

.75 

2 

58 

48920 

.65 

51092 

.72 

2 

59 

46552 

.69 

48489 

.75 

1 

59 

48959 

.65 

51135 

.72 

1 

60 

46594 

.69 

4-.-..1; 

.75 

0 

60 

48998 

.65 

51178 

.72 

0 

M. 

Cosine.  Dl" 

CtititllJ.'. 

Dl" 

M. 

M. 

Uorfne. 

Dl" 

Cotang. 

Dl" 

M. 

73° 


73= 


SINES  AND  TANGENTS. 


19° 


M.  i  Sine. 

Dl" 

Tang. 

1)1" 

M. 

M. 

Sine.  ;  Dl"   Tang. 

Dl" 

M. 

0 
1 

9.48998 
49037 

0.65 

9.51178 
51221 

0.72 

60 
59 

0 
1 

9.51264 
51301 

0.61 

9.53697 
53738 

0.68 

60 
59 

2 

49076 

.65 

51264 

.72 

58 

2 

51338 

.61 

A1 

53779 

.68 

AQ 

58 

3 

49115 

.65 

51306 

.71 

57 

3 

51374 

.0  J 

53820 

.00 

57 

4 

49153 

.65 

51349 

.71 

56 

4 

51411 

.61 
/>  i 

53861 

.68 

AQ 

56 

5 

49192 

.65 

51392 

.71 

55 

5 

51447 

.01 

53902 

.OO 

55 

6 

49231 

.64 

51435 

.71 

54 

6 

51484 

.61 

53943 

.68 

54 

7 
8 

49269 
49308 

.64 
.64 

RA 

51478 
51520 

.71 
.71 

53 
52 

7 
8 

51520 
51557 

.61 
.61 

A1 

53984 
54025 

.68 
.68 

AQ 

53 
52 

9 

49347 

.o4 

51563 

51 

9 

51593 

.01 

54065 

.Do 

51 

10 

49385 

.64 
/>  j 

51606 

.71 

50 

10 

51629 

.61 

Al 

54106 

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AQ 

50 

11 

9.49424 

.04 

9.51648 

49 

11 

9.51666 

.01 

A  A 

9.54147 

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AO 

49 

12 

49462 

.64 

51691 

.71 

48 

12 

51702 

.ou 

54187 

.00 

48 

13 

49500 

.64 

a  A 

51734 

.71 

*7l 

47 

13 

51738 

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AA 

54228 

.68 

AQ 

47 

14 

49539 

.04 

51776 

•  7  1 

46 

14 

51774 

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54269 

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46 

15 

49577 

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A/I 

51819 

.71 

45 

15 

51811 

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I'M 

54309 

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AQ 

45 

16 

49615 

.64 

51861 

.71 

44 

16 

51847 

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£i  A 

54350 

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AQ 

44 

17 

49654 

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51903 

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43 

17 

51883 

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54390 

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43 

18 
19 

49692 
49730 

.64 
.64 

51946 

51988 

.71 
.71 

42 
41 

18 
19 

51919 
51955 

.60 
.60 

54431 
54471 

.68 
.67 

42 
41 

20 

49768 

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52031 

.71 

40 

20 

51991 

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54512 

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40 

21 

9.49806 

.64 

9.52073 

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39 

21 

9.52027 

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9.54552 

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AT 

39 

22 

49844 

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/>*> 

52115 

•70 

38 

22 

52063 

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Art 

54593 

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A>7 

38 

23 
24 

49882 
49920 

.00 
.63 

52157 
52200 

•70 
.70 

37 
36 

23 
24 

52099 
52135 

.OU 

.60 

54633 
54673 

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37 
36 

25 

49958 

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52242 

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35 

25 

52171 

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54714 

.67 

35 

26 

49996 

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52284 

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34 

26 

52207 

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54754 

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34 

27 

50034 

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52326 

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54794 

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54835 

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54875 

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32 

50223 

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52536 

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52421 

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54995 

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33 

50261 

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52578 

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52456 

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55035 

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27 

34 

50298 

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52620 

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26 

34 

52492 

.5S 

55075 

.67 

26 

35 
36 

50336 
50374 

.63 
.63 

52661 
52703 

.70 
.70 

25 
24 

35 
36 

52527 
52563 

.55 
.59 

55115 
55155 

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25 
24 

37 

50411 

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52745 

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55195 

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55275 

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40 

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52870 

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9.52912 

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19 

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9.55355 

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19 

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50598 

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52953 

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42 

52775 

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55395 

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18 

43 

50635 

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52995 

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17 

43 

52811 

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55434 

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17 

44 

50673 

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AO 

53037 

.69 

16 

44 

52846 

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r  ft 

55474 

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A  A 

16 

45 

50710 

•  OJ 

53078 

.69 

15 

45 

52881 

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55514 

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15 

46 

50747 

.62 

53120 

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14 

46 

5291  6 

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55554 

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14 

47 

50784 

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53161 

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13 

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52951 

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55593 

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13 

48 

50821 

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53202 

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12 

48 

52986 

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55633 

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AA 

12 

49 

50858 

•  0.6 

53244 

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11 

49 

53021 

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55673 

.01 

11 

50 

50896 

.62 

53285 

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10 

50 

53056 

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55712 

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10 

51 

9.50933 

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9.53327 

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9 

51 

9.53092 

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9.55752 

.66 

9 

52 

50970 

.62 

53368 

.68 

8 

52 

53126 

.58 

55791 

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8 

53 

51007 

•62 

6C 

53409 

.69 

rtfl 

7 

53 

53161 

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C  Q 

55831 

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7 

54 

51043 

4 

53450 

.oV: 

6 

54 

53196 

.OO 

5587( 

.Of 

6 

55 

51080 

.61 

53492 

.6? 

5 

55 

53231 

.58 

55910 

.66 

5 

56 

51117 

.61 

A1 

53533 

.69 
Aft 

4 

56 

53266 

.58 

e  Q 

55949 

.66 

AA 

4 

57 

51154 

•  VI 

53574 

.Ot 

3 

57 

53301 

.UO 

55989 

.OO 

3 

58 

51191 

.61 

/*-! 

53615 

.69 

QQ 

2 

58 

53336 

.58 

CQ 

56028 

.66 

AA 

2 

59 

51227 

.OJ 

53656 

.OO 

1 

59 

53370 

.OO 

56067 

.01 

1 

60 

51264 

.61 

53697 

.68 

0 

60 

53405 

.58 

56107 

.66 

0 

M. 

Cosine. 

Dl" 

Cotsmgr.  Dl" 

M. 

M. 

Cosine,  j  Di" 

Cotang 

Dl"  j  M. 

71s 


37 


20° 


TABLE  IV.— LOGARITHMIC 


Al 

Sine.   Dl" 

Tang.   D," 

M, 

31. 

Sin*.    Dl" 

T»ug. 

Di'' 

M. 

0 

1 

9.53405 
53440 

0.58 

9.56107  1.  R. 
56146  U*^? 

60 
59 

0 
1 

9.55433 
55466 

0.55 

9.584  Ib 
58455 

0.63 

60 
59 

2 
3 

5M475 
53509 

!58 

CO 

56185  'JJ 
56224|  'X 

58 

57 

2 
3 

55499 
55532 

.55 
.55 

58493 
58531 

.63 

AO 

58 
57 

4 

53544 

.Do 

5.6264 

.Oil 

56 

4 

555C.4 

.55 

58569 

.OO 

56 

5 

53578 

.58 

56303 

.65 

55 

5 

55597 

.55 

58606 

.63 

55 

6 

53613 

.58 

CO 

56342 

.65 

A  c. 

54 

6 

55630 

.55 

C  C 

58644 

.63 

A*? 

54 

7 
8 

53647 
53682 

.Do 

.57 

57 

56381 
56420 

.00 

.65 

Afv 

53 

52 

7 
8 

55663 
55695 

•  DO 

.55 

58681 
58719 

»Otj 

.63 
/»•> 

53 
52 

9 

53716 

i 

56459 

.00 

51 

9 

55728 

.54 

58757 

.00 

51 

10 

53751 

.57 

56498 

.65 

50 

10 

55761 

.54 

58794 

.63 

50 

11 

9.53785 

.57 

9.56537 

.65 

49 

11 

9.55793 

.54 

9.58832 

.63 

49 

12 

53819 

.57 

56576 

.65 

48 

12 

55826 

.54 

58869 

.oz 

48 

13 

53854 

.57 

56615 

.65 

47 

13 

55858 

.54 

58907 

.62 

47 

14 

53888 

.57 

56654 

.65 

46 

14 

55891 

.54 

58944 

.62 

46 

15 

53922 

.67 

c  7 

56693 

.65 

A  C. 

45 

15 

55923 

.54 

58981 

.62 

AO 

45 

16 

53957 

,91 

C7 

56732 

.00 
A  ^ 

44 

16 

55956 

.54 

59019 

.OZ 
AO 

44 

17 

53991 

.0  1 

56771 

.OO 

43 

17 

55988 

.54 

59056 

.OZ 

43 

18 

54025 

.57 

56810 

.65 

42 

18 

56021 

•  54 

59094 

.62 

42 

19 

54059 

56849 

.65 

41 

19 

56053 

.54 

59131 

.62 

41 

20 

54093 

.57 

56887 

.65 

40 

20 

56085 

.54 

59168 

.62 

40 

21 

9.54127 

.57 

c  7 

9.56926 

.65 

A  ^ 

39 

21 

9.56118 

.54 

9.59205 

.62 

AO 

39 

22 

54161 

•91 

56965 

.OO 

38 

22 

56150 

.54 

59243 

.OZ 

38 

23 

54195 

.57 

57004 

.65 

37 

23 

56182 

.54 

59280 

.62 

37 

24 

54229 

.57 

57 

57042 

.64 

A/i 

36 

24 

56215 

.54 

59317 

.62 

AO 

36 

25 

54263 

1 

57081 

.04 

35 

25 

56247 

•  54 

59354 

.OZ 

35 

26 

54297 

.57 

57120 

.64 

34 

26 

56279 

•  54 

59391 

.62 

34 

27 

54331 

.56 

cc 

57158 

.64 

A  1 

33 

27 

56311 

.54 

59429 

.62 

AO 

33 

28 

54365 

•  OO 

KA 

57197 

.04 

AJ 

32 

28 

56343 

•54 

K.A 

59466 

.OZ 
AO 

32 

29 
30 

54399 
54433 

•  DO 

.56 

57235 
57274 

.04 

.64 

31 
30 

29 
30 

56375 
56408 

•  04 

•  53 

59503 
59540 

.Oz 

.62 

31 
30 

31 

9.54466 

•  Do 

9.57312 

.64 

29 

31 

9.56440 

•53 

9.59577 

.62 

29 

32 
33 

54500 
54534 

.56 

57351 
57389 

.64 

A/I 

28 
27 

32 
33 

56472 
56504 

.53 
.53 

CO 

59614 
59651 

.62 
.62 

AO 

28 
27 

34 
35 

54567 
54601 

.56 

57428 
57466 

.04 

.64 

26 
25 

34 
35 

5653C 
56568 

•  OO 

.53 

59688 
59725 

•02 

.62 

26 
25 

36 

54635 

.56 

57504 

.64 

24 

36 

56599 

.53 

59762 

.62 

24 

37 
38 

54668 
54702 

.56 

57543 
57581 

.64 
.64 

23 
22 

37 

38 

56631 
56663 

•53 
.53 

59799 
59835 

.61 

23 

22 

39 

54735 

.56 

57619 

.64 
.64 

21 

39 

56695 

.53 

CO 

59872 

.61 

Al 

21 

40 

54769 

57658 

20 

40 

56727 

•  DO 

59909 

.O  1 

20 

41 

9.54802 

.56 

r  £ 

9.57696 

.64 

19 

41 

9.56759 

.53 

9.59946 

.61 

19 

42 

54836 

.DO 

c  t> 

57734 

.64 

18 

42 

56790 

.53 

59983  '!!! 

18 

43 

54869 

.DO 

57772 

•  64 

17 

43 

56822 

•  53 

60019 

.01 

17 

44 

54903 

.56 
.56 

57810 

.64 

16 

44 

56854 

•  53 

c>> 

60056 

.61 

Al 

16 

45- 

54936 

57849 

.04 

15 

45 

56886 

•  DO 

60093 

•  01 

15 

46 

54969 

.56 

57887 

.64 

14 

46 

56917 

.53 

6013(1 

.61 

14 

47 

55003 

.55 
.55 

57925 

.63 

13 

47 

56949 

.53 

CO 

60166 

.61 
fii 

13 

48 
49 

55036 
55069 

.55 

K.Z. 

57963 
58001 

.63 

12 
11 

48 
49 

56980 
57012 

•  Do 

.53 

6020?, 
60240 

.01 

.61 

12 
11 

50 
51 

55102 
9.55136 

.00 

.55 

58039 
9.58077 

.63 

10 
9 

50 
51 

57044 
9.57075 

.53 
.53 

60276 
9.60313 

.61 
.61 

10 
9 

52 

55169 

.55 

r  r 

58115 

.68 

8 

52 

57107 

.52 

60349 

.61 

8 

53 

55202  '"? 

58153 

.63 

A  -J 

7 

53 

57138 

.52 

60386 

.61 

7 

54 

55235 

•  JO 

58191 

.00 

6 

54 

57109 

.52 

60422 

.61 

6 

55 

55268 

.55 

r  - 

58229 

.63 

5 

55 

57201 

.52 

60459 

.61 

5 

56 

55301 

•  5o 

f  r 

58267 

.63 

4 

56 

57232 

.52 

60495 

.61 

4 

57 

55334 

.Do 

58304 

.63 

3 

57 

57264 

.52 

60532 

.61 

3 

58 

55367 

.55 

r  r 

58342 

.63 

2 

58 

57295 

.52 

60568 

.61 

2 

59 
60 

55400 
55433 

.OD 

.55 

58380 
58418 

.63 
.63 

1 

0 

59 
60 

57326 
57358 

.52 
.52 

60605 
60641 

.61 
.61 

1 

0 

M. 

Cosine. 

D"l 

Ootaner.  Dl" 

M. 

M. 

Cosine. 

Dl" 

Cotang. 

Dl"  |  M. 

SINES  AND  TANGENTS, 


23° 


M.    Sine,  j  Di" 

Tung.   Dl" 

M. 

M. 

Sine.   1)1"   Tang,  j  D." 

31. 

o  y.:>7:;.~>s 

9.60641 

C,!) 

T 

9.5UloS  n  ,n  9.  62785  L  .,, 

60 

l 

57389 

0.52 

60677 

0.61 

A1 

59 

i 

59218  ;U'™ 

62820 

u.oy 

CO 

59 

2 

57420 

.oz 

60714 

.01 
C  1 

58 

2 

59247 

.49 

62855 

,08 
e,8 

58 

3 

57451 

ro 

60750 

.0  1 

/*  A 

57 

3 

59277 

62890 

.00 

CO 

57 

4 

57482 

q 

60786 

,OU 
en 

56 

4 

59307 

.49 

62926 

.08 

CO 

56 

5 

57514 

•J,   60823 

.  OU 

55 

5 

59336 

62961 

•  Oo 

55 

6 

57545 

•~   608.)9 

.60 

54 

6 

59366 

.49 

62996 

.58 

54 

«  7 

57576 

60895 

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An 

53 

7 

59396 

.49 

63031 

.58 

CO 

53 

8 
9 

57607 
57638 

'.52 

FLO 

61)9-51 
60967 

.OU 

.60 
fift 

52 
51 

8 
9 

59425 
59455 

^49 

63066 
63101 

.08 

.58 

eo 

52 
5] 

10 

57669 

•  OZ 
c  o 

61004 

.OU 
Aft 

50 

10 

59484 

j  63135 

.JO 
eo 

50 

11 

12 
13 
14 

9.57700 

57762 
57793 

.o/ 
.52 
.52 
.52 

r  1 

9.61040 
61076 
61112 
61148 

.OU 
.60 
.60 
.60 

AH 

49 
48 
47 
46 

11 
12 
13 
14 

9.59514 
59543 
59573 
59602 

!49 
.49 
.49 

9.63170 
63205 
63240 

63275 

.08 

.58 
.58 
.58 

eo 

49 

48 
47 
46 

15 

57824 

.01 

T  1 

61184 

.OU 

60 

45 

15 

59632 

'*?   6H310 

.08 

.58 

45 

16 

57855 

.0  1 

r  1 

61220 

An 

44 

16 

59661 

JJ  !  63345 

CO 

44 

17 

57885 

.51 

r  I 

61256 

.OU 
An 

43 

17 

59690 

A  U 

63379 

.08 
eo 

43 

18 

57916 

.01 

61292 

.OU 

42 

18 

59720 

•4iy 

63414 

.08 

CO 

42 

19 

57947 

el 

61328 

An 

41 

19 

59749 

* 

63449 

.08 
F.Q 

41 

20 

57978 

•  01 

T  I 

61364 

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60 

40 

20 

59778 

•49 

63484 

•  OO 

.58 

40 

21 

9.58008 

.0  i 

r  | 

9.61400 

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39 

21 

9.59808 

4Q 

9.63519 

CO 

39 

22 
23 
24 

58039 
ft  SI)  70- 
58101 

.01 
.51 
.51 

el 

61436 
61472 
61508 

.OU 
.60 

fid 

38 
37 
36 

22 
23 
24 

59837 
59866 
59895 

!49   6?55* 

.49 
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.08 

.58 

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RQ 

38 
37 
36 

25 

58131 

•  01 

61544 

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60 

35 

25 

59924 

1  63657 

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35 

26 

58162 

61579 

60 

34 

26 

59954 

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63692 

*?Q  34 

27 

58192 

61615 

33 

27 

59983 

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63726  '- 

33 

28 

58223 

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.51 

61651 

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32 

28 

60012 

»^rt7 

.48 

63761 

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32 

29 

58253 

61687 

31 

29 

60041 

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63796 

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31 

30 

58284 

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61722 

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30 

30 

60070 

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30 

31- 

9.58314 

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9.61758 

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29 

31 

9.60099 

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9.63865!  .0 

29 

32 

58345 

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61794 

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28 

32 

60128 

63899 

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28 

58375 

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61830 

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27 

33 

60157 

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(^7 

27 

34 

58406 

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61865 

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26 

34 

60186 

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63968 

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.57 

26 

35 
36 

58436 
58467 

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61901 
61936 

169 

F;Q 

25 
24 

35 

36 

60215 
60244 

.48 

64003 
64037 

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25 
24 

37 
38 

58497 
58527 

•  0  1 

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61972 

62008 

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23 
22 

37 
38 

60273 
60302 

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64072 
64106 

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23 

22 

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58557 

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en 

62043 

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C.Q 

21 

39 

603311 

64140 

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58588 

•  OU 

62079 

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20 

40 

60359 

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64175 

20 

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9.58618 

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9.62114 

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19 

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9.60388 

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48 

9.64209 

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19 

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58648 

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62150 

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18 

42 

60417 

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•48 

64243 

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.57 

18 

43 

58678 

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62185 

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17 

43 

60446 

64278 

17 

44 

58709 

en 

62221 

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16 

44 

60474 

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64312 

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57 

16 

45 

58739 

.  OU 

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15 

45 

60503 

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64346 

15 

46 

58769 

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62292 

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14 

46 

60532 

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64381 

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14 

47 

58799 

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en 

62327 

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13 

47 

60561 

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.48 

64415 

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13 

48 

58829 

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12 

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12 

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11 

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58889 

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51 

9.58919 

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51 

9.60675 

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9.64552 

.57 

e  7 

9 

52 

.58949 

.50 
en 

62504 

.Oi 
'er 

8 

52 

60704 

•4o 

64586 

.0  / 

.57 

8 

53 

58979 

•  O'J 
en 

62539 

.OV 
c  r 

7 

53 

60732 

40 

64620 

57 

7 

54 

59009 

.OU 
en 

62574 

.oy 

er 

6 

54 

60761 

•^to 
40 

64654 

.0  1 

.57 

6 

55 

59039 

.OU 

r  A 

62609 

.OV 

5 

55  i  60789 

•  rto 

4'  7 

64688 

^7 

5 

56 

59069 

.OU 

62645  '** 

4 

56    60818 

I 

64722 

.0  1 

4 

57 

59098 

.50 

e  A 

62680  1  •{• 

3 

57  1  60846 

.47 

64756 

.57 

1  P.7 

3 

58 

59128 

.OU 
CA 

62715 

e  r 

2 

58 

60875 

.47 

A7 

64790 

.91 

K7 

2 

59 

59158 

•  OU 

CA 

62750 

.oy 

r  n 

1 

59 

60903 

•  rt  ( 

64824 

•  0  1 

e.7 

1 

60 

59188 

.OU 

62785 

.Ov 

0 

60 

60931 

.47 

64858 

.O/ 

0 

M. 

Cosine.  |  PI" 

Cot  an  ar. 

Dl" 

M. 

M.   Cosine. 

Dl" 

Ootang. 

Dl" 

M. 

67C 


TABLE  IV.— LOGARITHMIC 


31. 

Sine.   Di" 

Tang. 

Dl"   31. 

M. 

.Sine. 

Dl" 

Tang. 

Dl" 

II. 

0 

1 

9.60931, 

609M  °-l; 

9.64858 
64892 

0.57 

60 
59 

0 

1 

9.62595 
62622 

0.45 

9.66867 
66900 

0.55 

60 
59 

2 

60988   ,1 

64926  'IL 

58 

2 

62649 

.45 

66933 

.55 

c  c 

58 

3 

61016 

.*< 

64960 

57 

3 

62676 

.45 

66966 

.00 

57 

4 

61045 

.47 

64994 

.57 

56 

4 

62703 

.45 

66999 

.55 

56 

5 

61073 

.47 

A  7 

65028 

.57 

t  7 

55 

5 

62730 

.45 

A  Z* 

67032 

.55 

t  c 

55 

6 

61101 

.4< 

65062 

.0  / 

54 

6 

62757 

.40 

67065 

.00 

54 

7 

61129 

.47 

A  7 

65096 

.56 

53 

7 

62784 

.45 

A  f^ 

67098 

.55 

c  r 

53 

8 
9 

61158 
61186 

.4< 
.47 

65130 
65164 

!56 

52 
51 

8 
9 

62811 
62838 

.40 

.45 

67131 
67163 

.00 

.55 

52 
51 

10 

61214 

.47 

65197 

.56 

50 

10 

62865 

.45 

67196 

.55 

50 

11 

9.61242 

.47 

9.65231 

.56 

49 

11 

9.62892 

.45 

9.67229 

.55 

49 

12 
13 

61270 
61298 

.47 
.47 

65265 
65299 

.56 
.56 

48 
47 

12 
13 

62918 
62945 

.45 
.45 

67262 
67295 

.55 
.55 

48 
47 

14 

61326 

.47 

47 

65333 

•s 

46 

14 

62972 

.45 

A  f\ 

67327 

.55 

46 

15 

61354 

.4  1 

65366  '"" 

45 

15 

62999 

•40 
4S 

67360 

R  Cv 

45 

16 

61382 

A  7 

65400  '2 

44 

16 

63026 

.40 

A  K. 

67393 

•  Ot> 

C  r 

44 

17 

61411 

.4* 

A  7 

65434   .„ 

43 

17 

63052 

.40 

A  R 

67426 

.00 

c  A 

43 

18 

61438 

.4< 

65467   _ 

42 

18 

63079 

.40 

67458 

.00 

42 

19 

61466 

.47 

A  *7 

65501 

-/» 

41 

19 

63106 

.45 

67491 

.54 

41 

20 

61494 

.4* 

65535 

.00 

40 

20 

63133 

.44 

67524 

: 

40 

21 

9.61522 

A  T 

9.65568  '™ 

39 

21 

9.63159 

•t!  |9.67556 

.54 

39 

22 

61550 

.47 

65602  '{[X 

38 

22 

63186 

•**   67589 

38 

23 

61578 

4fi 

65636  *2 

37 

23 

63213 

44 

67622 

Rl 

37 

24 
25 

61606 
61634 

.40 
.46 

A  A 

65669  '2 
65703  '2 

36 
35 

24 
25 

83239 
63266 

•44 

.44 

A  A 

67654 

67687 

.0-4 
.54 

f\A 

36 
35 

26 

61662 

.40 

4fi 

65736  '2 

34 

26 

63292 

.44 

A  A 

67719 

.04 

34 

27 

61689 

.40 

A  A 

65770 

n.  A 

33 

27 

63319 

.44 

A  A 

67752 

C  ,1 

33 

28 

61717 

.40 
A  A 

65803 

.00 

32 

28 

63345 

.44 

A  A 

67785 

.04 

32 

29 
30 

61745 
61773 

.40 
.46 

A  a 

65837 
65870 

^56 

FvA 

31 

30 

29 
30 

63372 
63398 

.44 

.44 

A  A 

67817 
67850 

.54 

31 
30 

31 
32 

9.61800  ' 
61828  !  "JJ 

9.65904 
65937 

.00 

.56 
56 

29 
28 

31 
32 

9.63425 
63451 

.44 

.44 

44 

9.67882 
67915 

.54 
.54 

29 

28 

33 

61856 

A  A 

65971 

•jJU 

27 

33 

63478 

.44 
A  1 

67947 

C  | 

27 

34 

61883 

.40 

4fi 

66004 

.56 

26 

34 

63504 

.44 
44 

67980 

.04 

.54 

26 

35 

61911 

•4O 

66038 

25 

35 

63531 

•  44 

68012 

25 

36 

61939 

.46 

66071 

fit* 

24 

36 

63557 

.44 

1  A 

68044 

.54 

KJ 

24 

37 

61966 

•46 

66104 

.00 

flfi. 

23 

37 

6S583 

.44 
44 

68077 

.04 

.54 

23 

38 

61994 

46, 

66138  i  '"" 

22 

38 

63610 

•  44 

A  A 

68109 

22 

39 
40 

62021 
62049 

.40 

.46 

661711  '2 
66204  '2 

21 

20 

39 
40 

63636 
63662 

.44 

.44 

68142 
68174 

.54 

21 

20 

41 

9.62076!  'JJ 

9.66238  TI 

19 

41 

9.63689 

.44 

9.68206 

.54 

19 

42 

62104  ' 

66271   - 

18 

42 

63715 

.44 

A  A 

68239 

.54 

18 

43 

62131 

66304  TT 

17 

43 

63741 

.44 

68271 

17 

44 

62159  ' 

66337  'rr 

16 

44 

63767 

.44 

A  A 

68303 

16 

45 

-62186!  1° 

66371  'J? 

15 

45 

63794 

.44 
44 

68336 

15 

46 

62214 

Af\ 

664041  '2 

14 

46 

63820 

•4^t 
44. 

68368 

" 

14 

47 

48 

62241 
62268 

.40 

.46 

A  A 

66437  '2 
66470  :2 

13 
12 

47 
48 

63846 
63872 

.44 

.44 

A  A 

68400 
68432 

1J  '* 

49 

62296 

.40 

66503  '2 

11 

49 

63898 

.44 

68465 

.04  !  i  , 

50 

62323 

.46 
4^ 

66537 

10 

50 

8*924 

.44 

49 

68497 

11  10 

51 

9.62350 

•40 
4i 

9.66570  '2 

9 

51 

9.63950 

.40 

4S 

9.68529 

M^ 

52 

62377 

.40 

66603  •?? 

8 

52 

63976 

•rt'> 

68561 

Q 
fid      " 

53 

62405 

'H 

66636  '?! 

7 

53 

64002 

.43 

A  *-J 

68593 

.04 

7 

54 

62432 

.4.) 

66669  '2 

6 

54 

64028 

.4o 

68626 

6 

55 

62459 

.45 

A  & 

66702  "2 

5 

55 

64054 

••}?!  68658 

.54 

5 

56 

62486 

.40 

66735  TI 

4 

56 

64080 

•JJ  i  68690 

4 

57 

62513 

.45 

66768   „ 

3 

57 

64106 

•*J   68722 

.54 

K.  O 

3 

58 

62541!  '•[? 

66801   *?? 

2 

58 

64132 

•J:?   68754  •£ 

2 

59 

62568  •** 

66834   .'- 

1 

59 

64158 

•JJ   68786 

.0.} 

1 

60 

62595  -4o 

66867 

0 

60 

64184 

68818 

.53 

0 

M. 

Cosine.  Dl" 

Cotang.  Dl" 

31. 

M. 

Cosine.   Dl"  Cotang.j  Dl"   M. 

SINES  AND  TANGENTS. 


27° 


M. 

Sine. 

Dl" 

Tang.   Dl" 

M. 

M. 

Sine. 

Dl" 

Tang. 

Di" 

M. 

0 
1 

9.64184 
64210 

..J  9.68818 
*   68850 

0.53 

CO 

60 
59 

0 
1 

9.65705 
65729 

0.41 

A  1 

9.70717 

70748 

0.52 

60 
59 

2 

64236 

•JJ   68882 

.Oo 

58 

2 

65754 

.41 

70779 

.52 

58 

3 
4 

64262 

64288 

.43 
.43 

68914 
68946 

.53 
.53 

57 
56 

3 
4 

65779 
65804 

.41 
.41 

70810 
70841 

.52 
.52 

57 
56 

5 

64313 

.43 

A  9 

68978 

.53 

CO 

55 

5 

65828 

.41 

A  1 

70873 

.52 

CO 

55 

6 

64339 

.4o 

69010 

•Oo 

54 

6 

65853 

.41 

70904 

.02 

54 

7 

64365 

.43 

69042 

.53 

CO 

53 

7 

65878 

.41 

A  1 

70935 

.52 

co 

53 

8 

64391 

.43 

,1  O 

69074 

.OO 

CO 

52 

8 

65902 

.41 

A  1 

70966 

.02 

en 

52 

9 

61417 

A6 

A  O. 

69106 

.00 

r  0 

51 

9 

65927 

.41 

70997 

•02 

CO 

51 

10 
11 
12 

64442 
9.64468 
64494 

.4o 
.43 
.43 

A  'i 

69138 
9.69170 
69202 

•  OO 

.53 
.53 

C  O 

50 
49 

48 

10 
11 

12 

65952 
9.65976 
66001 

.41 
.41 
.41 

A  1 

71028 
9.71059 
71090 

•  OZ 

.52 
.52 

CO 

50 
49 

48 

13 
14 

64519 
64545 

,4o 

.43 

A  9 

69234 
69266 

.Oo 
.53 

C  O 

47 
46 

13 
14 

66025 
66050 

•  41 

.41 

A  1 

71121 
71153 

.02 

.52 

CO 

47 
46 

15 

64571 

.4o 

A  °. 

69298 

.00 

CO 

45 

15 

66075 

.41 

A  1 

71184 

•02 

CO 

45 

16 
17 

64596 
64622 

.4o 

.43 

69329 
69361 

.00 

.53 

44 
43 

16 
17 

66099 
66124 

•41 

.41 

71215 
71246 

.OZ 

.52 

44 
43 

18 
19 

64647 
64673 

.43 
.43 

69393 
69425 

.53 
.53 

CO 

42 
41 

18 
19 

66148 
66173 

.41 
.41 

41 

71277 
71308 

.52 
.52 

42 
41 

20 
21 

64698 
9.64724 

.42 

69457 
9.69488 

.Do 

.53 

CO 

40 
39 

20 
21 

66197 
9.66221 

•41 

.41 
.41 

71339 
9.71370 

'.52 
.52 

40 
39 

22 

64749 

49 

69520 

.00 

CO 

38 

22 

66246 

41 

71401 

CO 

38 

23 

64775 

42 

69552 

.Do 

C  •) 

37 

23 

66270 

•rxl 

41 

71431 

•  OZ 

.52 

37 

24 

64800 

.^tz 

69584 

•  Do 

co 

36 

24 

66295 

.rt  1 

41 

71462 

CO 

36 

25 

26 

27 

64826 
64851 
64877 

.42 
.42 

69615 
69647 
69679 

.Do 

.53 
.53 

CO 

35 
34 
33 

25 

26 

27 

66319 
66,343 
66368 

.41 

.41 
.41 

41 

71493 
71524 
71555 

•  Oz 

.51 
.51 
ci 

35 
34 
33 

28 

64902 

49 

69710 

•  Do 

CO 

32 

28 

66392 

•rtl 

40 

71586 

.01 
Kl 

32 

29 

64927 

49 

69742 

.Do 

31 

29 

66416 

.^iv 

40 

71617 

•  OX 

C1 

31 

30 

64953 

•4Z 
49 

69774 

CQ 

30 

30 

66441 

•4U 

71648 

•  01 

30 

31 
32 
33 

9.64978 
65003 
65029 

•*±Z 

.42 

.42 
4') 

9.69805 
69837 
69868 

•  Do 

.53 

.53 

cq 

29 

28 
27 

31 

32 
33 

9.66465 
66489 
66513 

.'40 

.40 

40 

9.71679 
71709 
71740 

.51 

.51 

P.1 

29 

28 
27 

34 

65054 

A** 

.42 

69900 

.Do 

cq 

26 

34 

66537 

A" 
40 

71771 

•  01 

26 

35 

65079 

69932 

•  Do 

CO 

25 

35 

66562 

•rrv 

40 

71802 

M 

25 

36 
37 

65104 
65130 

.'42 

69963 
69995 

.Do 

.53 

24 
23 

36 
37 

66586 
66610 

.4v 

.40 

71833 
71863 

•  01 

.51 

r  1 

24 
23 

38 

65155 

49 

70026 

.53 

co 

22 

38 

66634 

40 

71894 

.oJ 

Cl 

22 

39 

65180 

42 

70058 

.Do 

CO 

21 

39 

66658 

•4" 

40 

71925 

•  0  J 

e.1 

21 

40 
41 
42 
43 

65205 
9.65230 
65255 
65281 

.*±Z 

.42 
.42 
.42 

70089 
9.70121 
70152 
70184 

•  OZ 

.52 
.52 
.52 

20 
19 
18 
17 

40 
41 
42 
43 

66682 
9.66706 
66731 
66755 

•rrV 

.40 
.40 
.40 

71955 
9.71986 
72017 
72048 

•  01 

.51 
.51 
.51 

20 
19 
18 
17 

44 

65306 

.42 

70215 

.52 

CO 

16 

44 

66779 

.40 
40 

72078 

.51 
.51 

16 

45 

65331 

70247 

.Oz 

15 

45 

66803 

Av 

72109 

15 

46 

65356 

.42 

49 

70278 

.52 

CO 

14 

46 

66827 

.40 
4ft 

72140 

.51 

14 

47 

65381 

.4z 

1  O 

70309 

.Oz 

13 

47 

66851 

.4U 

72170 

C  -I 

13 

48 

65406 

.4z 

70341 

.52 

12 

48 

66875 

.40 

72201 

.01 

12 

49 

65431 

.42 

70372 

.52 

CO 

11 

49 

66899 

.40 
.40 

72231 

.51 
KI 

11 

50 

65456 

A  •) 

70404 

.Oz 

10 

50 

66922 

72262 

.ul 

c  -i 

10 

51 
52 

9.65481 
65506 

AL 
.42 

.42 

9.70435 
70466 

.52 
.52 

£9 

9 
8 

51 
52 

9.66946 
66970 

.40 

.40 

9.72293 
72323 

.01 

.51 
.51 

9 

8 

53 

65531 

70498 

.OZ 

7 

53 

66994 

72354 

7 

54 

65556 

.42 

70529 

.52 

6 

54 

67018 

.40 

72384 

.51 

6 

55 

65580 

.41 

1  1 

70560 

.52 

5 

55 

67042 

A  A 

72415 

.61 

C  I 

5 

56 

65605 

.4  1 

A  1 

70592 

.52 

4 

56 

67066 

.4U 

A  A 

72445 

.01 

c  T 

4 

57 

58 

65630 
65655 

.41 

.41 

A  1 

70623 
70654 

.52 

3 
2 

57 

58 

67090 
67113 

.40 
.40 

A  A 

72476 
72506 

•  01 

.51 

K1 

3 
2 

59 

65680 

.4  1 

70685 

.52 

1 

59 

67137 

.40 

72537 

.01 

1 

60 

65705 

.41 

70717 

.52 

0 

60 

67161 

.40 

72567 

.51 

0 

M. 

Cosine.  |  Dl" 

CotHllg. 

Dl" 

M. 

M.   Cosine.  Dl" 

Cotang.  Dl" 

M. 

62° 


TABLE    IV.— LOGARITHMIC 


M. 

Sine. 

1)1" 

Tang. 

1)1" 

M. 

31.    Sine. 

Dl" 

Tang. 

1)1" 

0. 

0 
1 

2 

9.l>7161 
67185 

67208 

0.40 
.40 

9.72567 

72598 
72628 

0.51 
.51 

K-l 

60 
59 

58 

0 

1 
2 

9.68557 
68580 
68603 

0.38 
.38 

QQ 

9.74375 

744D5 
74485 

0.50 
.50 

fiO 
59 

58 

3 

67232 

on 

72659 

.01 

C  -I 

57 

3 

68625 

.GO 
Oc 

7440,3 

r  f  , 

57 

4 
5 

67256 
67280 

•ov 

.39 

qq 

72689 
72720 

.0  I 
.51 
RI 

56 
55 

4 

5 

68648 
68671 

•  Go 

.38 
.38 

74494 
74524 

•  OU 

.50 
.50 

56 
55 

6 

67303 

•oy 

72750 

•  Ol 

r  •] 

54 

6 

68694 

oo 

74554 

r  f\ 

54 

7 

67327 

.39 

72780 

.01 

53 

7 

68716 

.00 
qo 

74583 

.OU 

C  A 

53 

8 

67350 

•  39 

72811 

52 

8 

68739 

.OO 

7461.°, 

.OU 

52 

9 

67374 

•  39 

72841 

-  .51 

51 

9 

68762 

.38 

OQ 

74643 

.50 

51 

10 

67398 

•  39 

qq 

72872 

.51 
si 

50 

10 

68784 

.00 

.38 

74673 

'49 

50 

11 

9.67421 

•oy 

9.72902 

.01 

49 

11 

9.68807 

°>s 

9.74702 

* 

49 

12 

67445 

•oy 

72932 

si 

48 

12 

68829 

•  oo 

.38 

74732 

"do 
.49 

48 

13 

67468 

•39 

72963 

•  O  1 

47 

13 

68S52 

74762 

47 

14 

67492 

•  39 

72993 

.51 

C  -| 

46 

14 

68875 

.38 

<>Q 

74791 

•49 

46 

15 

67515 

•39 

73023 

.01 

50 

45 

15 

68897 

.OO 

.38 

748-21 

•49 
4.0 

45 

16 
17 

18 

67539 
67562 
67586 

•39 
•39 

on 

73054 
73084 
73114 

!so 

.50 

SO 

44 

43 

42 

16 
17 

18 

68920 
68942 
68965 

.38 
.38 
.38 

74851 

74880 
74910 

.^±V' 

•49 
.49 
/in 

44 
43 
42 

19 

67609 

•ft)  9 

73144 

.OU 

41 

19 

68987 

74939  ' 

41 

20 

67C33 

.39 

73175 

.50 

40 

20 

69010 

.37 

37 

74969  '  ' 

40 

21 

9.67656 

.39 

9.73205 

.50 

39 

21 

9.69032 

i 
07 

9.74998 

39 

22 
23 

67680 
-  67703 

.39 
.39 

73235 
73265 

.50 

.50 

38 
37 

22 
23 

69055 
69077 

•  Ol 

.37 

O  7 

75028 
75058 

.49 
.49 

38 
37 

24 

67726 

.39 

73295 

•  .50 

36 

24 

69100 

•  o4 

07 

75087 

.49 

36 

25 

26 

67750 
67773 

.39 

.39 

73326 
73356 

.50 
.50 

35 
34 

25 
26 

69122 
69144 

•  O/ 

.37 

75117 

75146 

.49 

•49 

35 
34 

27 

67796 

.39 

73386 

.50 

33 

27 

69167 

i 

75176 

•49 

33 

28 

67820 

.39 

73416 

.50 

32 

28 

69189 

•3i 

75205 

•49 

32 

29 

67843 

.39 

73446 

.50 

31 

29 

69212 

o  7 

75235 

.49 

31 

30 

67866 

.39 

73476 

.50 

30 

30 

69234 

•  •  >  / 
O*7 

75264 

.49 

30 

31 

9.67890 

.39 

9.73507 

.50 

CA 

29 

31 

9.69256 

•61 

•  37 

9.75294 

.49 

29 

32 
33 

67913 
67936 

.39 

73537 
73567 

.OU 

.50 

28 
27 

32 
33 

69279 
69301 

.37 

75323 
75353 

.49 

28 
27 

34 
35 
36 

67959 
67982 
68006 

.39 
.39 
.39 

73597 
73627 
73657 

.50 
.50 
.50 

26 
25 
24 

34 

35 
36 

69323 
69345 
69368 

r37 

•37 

q7 

75382 
75411 
75441 

.41) 
.49 

.49 

26 
25 

24 

37 

68029 

.39 

73687 

.50 

23 

37 

69390 

»O  4 

07 

75470 

•49 

23 

38 

68052 

.39 

73717 

.50 

C  A 

22 

38 

69412 

•  6i 
•  37 

75500 

.49 

4Q 

22 

39 

68075 

•39 

73747 

.OU 

21 

39 

69434 

75529 

•4y 

21 

40 

41 

68098 
9.68121 

!:',8 

73777 
9.73807 

.50 
.50 

20 
19 

40 
41 

69456 
9.69479 

•  37 
.37 

q7 

75558 
9.75588 

.49 
.49 

20* 
19 

42 

68144 

•g®   73837 

.50 

18 

42    69501 

•  o  t 

37 

75617 

.49 

18 

43 

68167 

73867 

.50 

17 

43 

69523 

i 

75647 

.49 

17 

44 

68190 

.38 

73897 

.50 

16 

44 

69545 

07 

75676 

•49 

16 

45 

68213 

.38 

qo 

73927 

.50 
so 

15 

45 

69567 

«o< 

•  37 

75705 

•49 

15 

46 
47 

68237 
68260 

•  oo 

.38 

73957 
73987 

.OU 

.50 

14 
13 

46 
47 

69589 
69611 

.37 

75735 
75764 

^49 

14 
13 

48 

68283 

•38 

74017 

.50 

12 

48 

69633 

•37 

37 

75793 

.49 

12 

49 

68305 

•38 

74047 

.50 

11 

49 

69655 

i 
*>7 

75822 

.49  -, 

50 

68328 

.38 

74077 

.50 

10 

50 

69i>77 

07 

75852 

.49 

10 

51 

9.68351 

.38 

9.74107 

.50 

9 

51 

9.69699 

"o-  9.75881 

.49 

9 

52 

68374 

.38 

741E7 

.50 

8 

52 

69721 

!s7   7591° 

.49 

8 

53 

68397 

.38 

74166 

.50 

7 

53 

69743 

07 

75939 

•49 

7 

54 

68420 

.38 

74  1  '.16 

.50 

6 

54 

69765 

•Ol 

75969 

.49 

6 

55 

68443 

.38 

qo 

74226 

.50 

5 

55 

697S7 

•37 
9,7 

75998 

.49 

5 

56 

68466 

•  oo 

742.36 

.OU 

4 

56 

69809 

•O  4 

76027 

•4y 

4 

57 

68489 

.38 

74286 

.50 

3 

57 

69831 

.37 

76056 

.49 

3 

58 

68512 

.38 

74316 

.50 

2 

58 

69853 

•37 

3- 

76086 

.49 

2 

59 

68534 

.38 

74345 

.50 

1 

59 

69875 

< 

76115 

.49 

1 

60 

68557 

.38 

74375 

.50 

0 

60 

69897 

.36 

76144 

.49 

0 

M. 

Cosine. 

Dl" 

Cotang. 

Dl" 

M. 

M. 

Cosine. 

Dl" 

CntaiiLr. 

Dl" 

M. 

6O° 


SINES  AND  TANGENTS. 


31° 


M.    Sine. 

Dl" 

Taiig. 

Di" 

11. 

M. 

Sine'. 

1)1" 

Taiig. 

Dl" 

M. 

0 
1 

9.69897 
69919 

0.3(5 

0  /» 

9.76144 
76173 

0.49 

A  O 

60 
59 

0 
1 

9.71184 
71205 

0.35 

o  r 

9.77877 
77906 

0.48 

A  Q 

60 

59 

2 

69911 

.00 

76202 

.4l,» 

58 

2 

71226 

.OO 

77935 

.48 

58 

3 

69963 

.36 
«>/» 

76231 

.49 

57 

3 

71247 

.35 

3x 

77963 

.48 

A  Q 

57 

4 

69984 

.00 

76261 

.49 

56 

4 

71268 

0 

77992 

.48 

56 

5 
6 

70006 

70028 

.36 
.36 

76290 
7631  9 

.49 

.49 

55 
54 

5 
6 

71289 
71310 

.35 
.35 

78020 
78049 

.48 
.48 

55 
54 

7 
8 
9 

70050 
70072 
70093 

.36 
.36 
.36 

76348 
76377 
76406 

.48 

.48 

53 
52 
51 

7 
8 
9 

71331 
71352 
71373 

.35 
.35 
.35 

78077 
78106 
78135 

.48 
.48 

.48 

53 
52 
51 

10 

70115 

.36 

no 

76435 

.48 

A  Q 

50 

10 

71393 

.35 

0  C 

78  i  63 

.48 

A  U 

50 

11 

9.70137 

.OO 
°ifi 

9.76464 

•4o 
40 

49 

11 

9.71414 

.00 

.35 

9.78192 

.48 

.48 

49 

12 

70159 

..DO 

76493 

•  4o 

48 

12 

71435 

78220 

48 

13 

70180 

.36 

76522 

.48 

4Q 

47 

13 

71456* 

9C 

78249 

-.48 

47 

47 

14 
15 
16 

70202 
70224 
70245 

.36 
.36 

76551 

76580 
76609 

•4o 

.48 
.48 

4U 

46 
45 
44 

14 
15 
16 

-  71477 
71498 
71519 

.0  j 
.35 
.35 
.35 

78277 
78306 
78334 

.^t  < 
.47 
.47 
.47 

46 
45 
44 

17 

70267 

°.fi, 

76639 

•4o 

43 

17 

71539 

or 

78363 

A  7 

43 

18 

70288 

.00 

76668 

4k 

42 

18 

71560 

•  O  J 

.35 

78391 

•  !  . 

.47 

42 

19 

70310 

.00 

76697 

•4o 

41 

19 

71581 

78419 

41 

20 

70332 

.36 

Of> 

76725 

.48 

4Q 

40 

20 

71602 

.35 

OF, 

78448 

.47 

40 

21 

9.70353 

.00 
oc 

9.76754 

.48 

AQ 

39 

21 

9.71622 

•oo 
.35 

9.78476 

.47 

39 

22 

70375 

.00 

76783 

.'iO 

38 

22 

71643 

78505 

38 

23 

70396 

.36 

O/; 

76812 

.48 

A  Q 

37 

23 

71664 

.35 

78533 

.47 

37 

24 

70418 

.OO 

Q  A 

76841 

,4o 

A  Q 

36 

24 

71685 

°.4 

78562 

A  ^ 

36 

25 

70439 

.OO 

Ofi 

76870 

•4o 

A  Q 

35 

25 

71705 

•  o4 
04 

78590 

.4  1 

35 

26 

70461 

.OO 

Ofi 

76899 

,4o 
40 

34 

26 

71726 

•  O*± 
04 

78618 

A7 

34 

27 
28 

70482 
70504 

•  OO 

.36 

Of* 

76928 
76957 

•4:0 

.48 

33 

32 

27 
28 

71747 
71767 

•  O^r 

.34 

78647 
78675 

^47 

A  7 

33 

32 

29 

70525 

.00 

Ofi 

76986 

.48 
4ft 

31 

29 

71788 

•** 

78704 

•41 

31 

30 

70547 

.00 

Of 

77015 

*4o 
A  Q 

30 

30 

71809 

78732 

47 

30 

31 

9.70568 

.00 

Ofl 

9.77044 

.48 

10 

29 

31 

9.71829 

9.78760 

•4  4 

47 

29 

32 
33 
34 

70590 
70611 
70633 

.OO 

.36 
.36 

O  A 

77073 
77101 
77130 

.48 

.48 
.48 

A  Q 

28 
27 
26 

32 
33 
34 

71850 
71870 
71891 

'.34 
.34 

0,1 

78789 
78817 
78845 

•  !  t 

.47 
.47 

4  fr 

28 
27 
26 

35 

70654 

.00 

OA 

77159 

.4o 

40 

25 

35 

71911 

..14 

.34 

78874 

•  4* 

47 

25 

36 

o  7 

70675 
70697 

.00 

.36 

Ofi, 

77188 
77217 

•'io 

.48 
4ft 

24 
23 

36 
37 

71932 
71952 

.'34 
04 

78902 
78930 

•  4:  i 

.47 

24 
23 

38 

70718 

«OO 

77246 

.48 

22 

38 

71973 

•  OT: 
.34 

78959 

j" 

22 

39 

70739 

.00 

.  77274 

.48 

21 

39 

71994 

78987 

.4  / 

21 

40 

70761 

.36 

0  C 

77303 

.48 

A  Q 

20 

40 

72014 

O  1 

79015 

.47 

20 

41 

9.70782 

.00 

9.77332 

.48 

19 

41 

9.72034 

.o4 

9.79043 

7 

19 

42 
43 

70803 
70824 

.35 
.35 

•3  ^ 

77361 
77390 

.48 
.48 

4ft 

18 
17 

42 
43 

72055 
72075 

.34 
.34 

79072 
79100 

.47 
.47 

18 

17 

44 

70846 

..30 
ox 

77418 

.rro 
A  Q 

16 

44 

72096 

'[. 

79128 

47 

16 

45 

70867 

.OO 

77447 

.48 

4ft 

15 

45 

72116 

79156 

.4  1 
47 

15 

46 

70888 

"v 

77476 

.48 

14 

46 

72137 

.34 

79185 

.4  1 

•47 

14 

47 

70909 

0  ~ 

77505 

A  Q 

13 

47 

72157 

o  A 

79213 

47 

13 

48 
49 

70931 
70952 

.03 

.35 

77533 
77562 

.48 

.48 

12 
11 

48 
49 

72177 
72198 

.o4 
.34 

79241 
79269 

I 

.47 

12 
11 

50 
51 

70973 
9.70994 

.35 
.35 

77591 
9.77619 

.48 
.48 

10 
9 

50 
51 

72218 
9.72238 

.34 
.34 
.34 

79297 
9.79326 

.47 
.47 
.47 

10 
9 

52 

71015 

.OU 

77648 

.48 

8 

52 

72259 

79354 

8 

53 
54 
55 

71036 

71058 
71079 

.35 
.35 
.35 

77677 
77706 

77734 

.48 
.48 

.48 

7 
6 
5 

53 
54 
55 

72279 
72299 
72320 

.34 
.34 
.34 

79382 
79410 
79438 

.47 
.47 

.47 

7 
6 
5 

56 

71100 

.35 

77763 

.48 

4 

56 

72340 

•34 

79466 

.47 

4 

57 

71121 

.35 

77791 

.48 

3 

57 

72360 

.34 

79495 

.47 

3 

58 

71142 

.35 

0  r 

77820 

.48 

A  Q 

2 

58 

72381 

.34 

0/1 

79523 

.47 

2 

59 

71163 

.OD 

77849 

.48 

] 

59 

72401 

.o4 

79551 

.47 

1 

60 

71184 

.35 

77877 

.48 

0 

60 

72421 

.34 

79579 

.47 

0 

Mb 

Cosine. 

Dl" 

Cotans. 

Dl" 

M. 

31. 

Cosine. 

Dl" 

Cotanji. 

Dl" 

M. 

59° 


43 


58° 


32° 


TABLE  IV.— LOGARITHMIC 


38° 


M. 

Sine. 

Dl" 

Tang. 

Dl" 

M. 

M. 

Sine. 

Dl" 

Tang. 

Dl" 

M. 

0 
1 

9.72421 
72441 

0.34 

9.79579 
79607 

0.47 

60 
59 

0 

1 

9.73611 
73630 

0.32 

9.81252 
81279 

0.46 

60 

59 

2 

72461  j  ft 

79635 

.47 

58 

2 

73650 

.32 

QO 

81307 

.46 

58 

a 

4 

72482  •** 

72502   „* 

79663 
79691 

!47 

57 
56 

3 

4 

73669 
73689 

•  62 
.32 

81335 
81362 

.46 

.46 

57 
56 

5 

72522 

••'4 

79719 

.47 

55 

5 

73708 

.32 

81390 

.46 

55 

6 

72542 

.34 

79747 

.47 

54 

6 

73727 

.32 

81418 

.46 

54 

7 

72562 

.34 

79776 

.47 

53 

7 

73747 

.32 

81445 

.46 

53 

8 

72582 

.34 

79804 

.47 

52 

8 

73766 

.32 

81473 

.46 

52 

9 

72602 

.34 

79832 

.47 

51 

9 

73785 

.32 

81500 

.46 

51 

10 

72622 

.33 

79860 

.47 

50 

10 

73805 

•  32 

81528 

.46 

50 

11 

9.72643 

.33 

9.79888 

.47 

49 

11 

9.73824 

.32 

9.81556 

.46 

49 

12 

72663 

.33 

79916 

.47 

48 

12 

73843 

.32 

81583 

.46 

•48 

13* 
14 

72683 
72703 

.33 
.33 

79944 
79972 

.47 

.47 

47 
46 

13 
14 

73863 
73882 

.32 
.32 

81611 

81638 

.46 
.46 

47 
46 

15 

16 
17 

72723 
72743 
72763 

.33 
.33 
.33 

qq 

80000 
80028 
80056 

.47 
.47 
.47 

4,7 

45 
44 
43 

15 
16 
17 

73901 
73921 
73940 

.32 
.32 
.32 

QO 

81666 
81693 
81721 

.46 
.46 
.46 

AR 

45 
44 
43 

18 

72783 

•  60 

OQ 

80084 

.4  f 

42 

18 

73959 

•  62 

81748 

.40 

42 

19 
20 

72803 
72823 

•  66 
•33 

80112 
80140 

.47 

.47 

41 

40 

19 
20 

73978 
73997 

.32 
.32 

81776 
81803 

.46 
.46 

41 
40 

21 
22 

23 

9.72843 
72863 
72883 

•  33 
.33 
.33 

9.80168 
80195 
80223 

!47 
.47 

39 
38 
37 

21 
22 
23 

9.74017 
74036 
74055 

.32 
.32 
.32 

9.81831 

81858 
81886 

.46 
.46 
.46 

39 
38 
37 

24 

25 

72902 
72922 

*33 
.33 

00 

80251 
80279 

.47 
.47 

4-» 

36 
35 

24 

•25 

74074 
74093 

.32 
.32 

OO 

81913 
81941 

.46 

.46 

36 
35 

26 

72942 

•OO 
oo 

80307 

7 

34 

26 

74113 

•  OZ 

QO 

81968 

A  A 

34 

27 
28 

72962 
72982 

•OO 

.33 

80335 
80363 

!46 

33 

32 

27 

28 

74132 
74151 

•02 

.32 

81996 
82023 

•40 

.46 

33 
32 

29 
30 
31 
32 
33 

73002 
73022 
9.73041 
73061 
7303  1 

O  CO  CO  CO  CO  « 

o  co  co  eo  co  < 

80391 
80419 
9.80447 

80474 
80502 

.46 
.46 
.46 
.46 
.46 

31 
30 
29 
28 
27 

29 
30 
31 

32 
33 

74170 
74189 
9.74208 
74227 
74246 

•32 
.32 
.32 
.32 
.32 

82051 
82078 
9.82106 
82133 
82161 

.46 
.46 
.46 
.46 
.46 

31 
30 
29 

28 
27 

34 

73101 

•  66 

80530 

.46 

26 

34 

74265 

•32 

82188 

.46 

26 

35 

73121 

•  33 

80558 

.46 

25 

35 

74284 

•32 

82215 

.46 

25 

36 

73140 

•33 

80586 

.46 

24 

36 

74303 

•32 

82243 

.46 

24 

37 

73160 

.33 

QQ 

80614 

.46 

A  £ 

23 

37 

74322 

.32 

QO 

82270 

.46 

A  A 

23 

38 

73180 

•OO 
OQ 

80642 

.4o 

22 

38 

74341 

•62 
oo 

82298 

•4o 

22 

39 

73200 

•66 

80669 

.46 

21 

39 

74360 

•  62 

82325 

.46 

21 

40 

73219 

•33 

80697 

.46 

20 

40 

74379 

.32 

oo 

82352 

.46 

20 

41 

9.73239 

-33 
qq 

9.80725 

.46 

A  a 

19 

41 

9.74398 

•62 

qo 

9.82380 

.46 

AO 

19 

42 

73259 

•oo 

OQ 

80753 

.40 
A  a 

18 

42 

74417 

•62 

QO 

82407 

.40 

A  A 

18 

43 

73278 

•  oo 

80781 

.46 

17 

43 

74436 

.64 

82435 

.40 

17 

44 

73298 

•33 

OQ 

80808 

.46 

16 

44 

74455 

.32 

oo 

82462 

.40 

16 

45 

73318 

•66 

80836  '*° 

15 

45 

74474 

.62 

82489 

.46 

15 

46 

73337 

•33 

80864 

14 

46 

74493  '%* 

82517 

.46 

14 

47 

73357 

«33 

80892 

.46 

13 

47 

74512  •** 

82544 

.46 

13 

48 

73377 

•  33 

QQ 

80919 

.46 

A  -• 

12 

48 

74531 

•  6i 

O1 

82571 

.46 
i  A 

12 

49 

73396 

•OO 
OQ 

80947 

.4} 

A  t* 

11 

49 

74549 

•31 

82599 

.40 

11 

50 

73416 

•66 

QQ 

80975 

.46 

A  a 

10 

50 

74568  'Jf 

82626 

.46 

10 

51 

9.73435 

•66 

9.81003 

.4o 

9 

51 

9.74587  ••} 

9.82653 

.46 

9 

52 

73455 

•33 

OQ 

81030 

.46 

8 

52 

74606  •»} 

82681 

.46 

8 

53 

73474 

•00 

81058 

.46 

7 

53 

74625  •*} 

82708 

.45 

7 

54 

73494 

•  33 

QQ 

81086 

.46 

A  & 

6 

54 

74644  •*{ 

82735 

.45 

A  EL 

6 

55 

73513 

«oo 

81113 

.40 

5 

55 

74662  !  •*} 

82762 

.40 

5 

56 

73533 

•33 

81141 

.46 

4 

56 

74681  •*} 

82790 

.45 

4 

57 

73552 

•32 

oo 

81169 

.46 

3 

57 

74700  •{] 

82817 

.45 

3 

58 

73572 

•62 

81196 

.46 

2 

58 

74719  ^; 

82844 

.45 

2 

59 

73591 

•32 

oo 

81224 

.46 

1 

59 

74737  •*} 

82871 

.45 

1 

60 

73611 

•62 

81252 

.46 

0 

60 

74756)  ' 

82899 

.45 

0 

M. 

Cosine. 

Dl" 

Cotang. 

Dl" 

M. 

M. 

Cosine. 

Dl" 

Cotang. 

Dl" 

M. 

57° 


44 


56° 


SINES  AND    TANGENTS. 


M. 

Sine. 

Dl" 

Tang. 

IK" 

M. 

M. 

bine. 

Dl" 

Tung. 

Dl" 

M. 

0 
1 

9.74756 
74775 

0.31 

9.82899 
82926 

0.45 

60 

59 

0 
1 

9.75859 
75877 

0.30 

9.84523 
84550 

0.45 

60 
59 

2 

74794 

.31 

82953 

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58 

2 

75895 

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84576 

.45 
.1  - 

58 

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74812 

.31 

82980 

.45 

57 

3 

75913 

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84603 

.40 

57 

4 

74831 

.31 

83008 

.45 

56 

4 

75931 

.30 

84630 

.45 

56 

5 

74850 

.31 

83035 

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55 

5 

75949 

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84657 

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55 

6 

74868 

.31 

83062 

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54 

6 

75967 

.30 

84684 

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54 

7 

74887 

.31 

83089 

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53 

7 

75985 

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8471  1 

.45 

53 

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74906 

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Q1 

83117 

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A  C 

52 

8 

76003 

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0  A 

84738 

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52 

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74924 

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83144 

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76021 

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74943 

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83171 

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A  E*. 

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84791 

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83225 

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A  *\ 

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.45 
.45 

47 
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44 

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76146 

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84952 

.45 

44 

17 

75073 

.31 

83361 

.45 

43 

17 

76164 

.30 

84979 

.45 

43 

18 

75091 

.31 

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76182 

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85006 

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42 

19 

75110 

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75128 

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«  A 

85059 

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40 

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76342 

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85247 

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on 

85273 

.45 

32 

29 

75294 

•  Ol 

83686 

31 

29 

76378 

.0" 

85300 

31 

30 
31 
32 

75313 
9.75331 
75350 

.31 
.31 
.31 

Q1 

83713 
9.83740 
83768 

.45 

.45 
.45 

30 

29 

28 

30 
31 
32 

76395 
9.76413 
76431 

.30 

.30 
.29 

oft 

85327 
9.85354 

85380 

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.45 
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30 
29 

28 

33 

75368 

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O  1 

83795 

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76448 

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O  1 

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85434 

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75405 

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q  1 

83849 

A* 

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76484 

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85460 

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75423 

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O  1 

83876 

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24 

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83903 

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23 

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76519 

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85514 

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76537 

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85540 

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22 

39 

75478 

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83957 

21 

39 

76554 

.^y 

85567 

21 

40 
41 

75496 
9.75514 

.30 
.30 

on 

83984 
9.84011 

.45 
.45 
.45 

20 
19 

40 
41 

76572 
9.76590 

.29 
.29 
29 

85594 
9.85620 

.44 
.44 
.44 

20 
19 

42 

75533 

•  OU 

on 

84038 

A  £ 

18 

42 

76607 

on 

85647 

AA. 

18 

43 
44 

75551 
75569 

.60 

.30 

OA 

84065 
84092 

.40 

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4^ 

17 
16 

43 
44 

76625 
76642 

.zy 

.29 

85674 
85700 

.44 

.44 
.44 

17 

16 

45 

46 

75587 
75605 

.60 

.30 

84119 

84146 

•rtO 

.45 

15 
14 

45 
46 

76660 
76677 

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85727 
85754 

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84173 

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85780 

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A  f\ 

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9.76765 

n 

9.85887 

.44 
A  A 

9 

52 

75714 

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84307 

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76782 

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85913 

.44 

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8 

53 

75733 

.OU 

O  A 

84334 

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7 

53 

76800 

Q 

85940 

A  A 

7 

54 
55 
56 

75751 
75769 

75787 

.oU 
.30 
.30 
on 

84361 
84388 
84415 

.40 

.45 
.45 

A  P* 

6 
5 
4 

54 
55 

56 

76817 
76835 

76852 

.29 
.29 

85967 
85993 
86020 

.44 
.44 
.44 

A  A 

6 

5 
4 

57 
58 
59 

75805 
75823 
75841 

.oU 
.30 
.30 

84442 
84469 
84496 

*40 

.45 

.45 

3 
2 
1 

57 
58 
59 

76870 
76887 
76904 

.29 
.29 

86046 
86073 
86100 

.44 
.44 

.44 

3 
2 
1 

60 

75859 

.30 

84523 

.45 

0 

60 

76922 

.29 

86126 

.44 

0 

M. 

Cosine. 

Dl" 

Cotang.!  Dl" 

M. 

M. 

Cosine. 

Dl" 

Cotang. 

Dl" 

M. 

55C 


54' 


36= 


TABLE  IV.— LOGARITHMIC 


37° 


31. 

Sine. 

Di" 

Tang. 

Dl" 

31. 

31. 

Sine. 

Dl" 

Tang. 

Dl"   31. 

0 
1 

9.76922 
76939 

0.29 

9.86126 
86153 

0.44 

60 

59 

0 
1 

9.77946 
7796:; 

0.28 

9.87711 

877:18 

"-  '! 

2 

STB957 

.29 

86179 

.44 

58 

2 

77980 

,2s 

877114 

3 

76974 

.29 

86206 

.44 

57 

3 

77997 

.28 

87790 

'11  57 

4 

76991 

.29 

86232 

.44 

56 

4 

78013 

.28 

87817 

11  56 

5 

77009 

.29 

86259 

.44 

55 

5 

78030 

.28 

87843 

.44   ,  - 

6 

77026 

.29 

86285 

.44 

54 

6 

78047 

.28 

87869 

.44 

54 

7 

77043 

.29 

86312 

.44 

53 

7 

78063 

.2S 

87895 

.44 

53 

8 

77061 

.29 

86338 

.44 

52 

8 

78080 

.2s 

87922 

.44 

52 

9 

77078 

.29 

86365 

.44 

51 

9 

78097 

.28 

8794S 

.44 

51 

10 

77095 

2r\ 

86392 

.44 

50 

10 

78113 

.28 

87974 

.44 

50 

11 

9.77112 

9 

9.86418 

.44 

49 

11 

9.78130 

.Zo 

9.88000 

.44 

49 

12 

77130 

.29 

86445 

.44 

48 

12 

78147 

.28 

88027 

.44 

48 

13 

77147 

.29 

86471 

.44 

47 

13 

78163 

.28 

88053 

.44 

47 

14 
15 

77164 
77181 

.29 
.29 

86498 
8»524 

.44 
.44 

46 
45 

14 
15 

78180 
78197 

.28 
.28 

88079 
88105 

.44 
.44 

46 
45 

16 

77199 

.29 

86551 

.44 

44 

16 

78213 

.28 

88131 

.44 

44 

17 

77216 

•29 

86577 

.44 

43 

17 

78230 

.28 

88158 

.44 

43 

18 

77233 

.29 

86603 

.44 

42 

18 

78246 

.28 

88184 

.44 

42 

19 

77250 

.29 

86630 

.44 

41 

19 

78263 

.2s 

88210 

.44 

41 

20 
21 

77268 
9.77285 

•  29 
.29 

86656 
9.86683 

.44 
.44 

40 
39 

20 
21 

782SO 
9.78296 

.28 
.28 

88236 
9.88262 

.44 
.44 

40 
39 

22 

77302 

•29 

86709 

.44 

38 

22 

78313 

.28 

88289 

.44 

38 

23 

77319 

•29 

86736 

.44 

37 

23 

78329 

.28 

88315 

.4-4 

37 

24 

77336 

•29 

86762 

.44 

36 

24- 

78346 

.28 

88341 

.44 

36 

25 

26 

77353 

77370 

•29 
•29 

86789 
86815 

.44 

.44 

35 
34 

25 
26 

78362 
78379 

.28 
.28 

88367 
88393 

.44 
.44 

35 
34 

27 

77387 

•29 

86842 

.44 

33 

27 

78395 

.27 

88420 

.44 

32 

28 

77405 

•28 

86868 

.44 

32 

28 

78412 

.27 

88446 

.44 

32 

29 

77422 

•28 

86894 

.44 

31 

29 

78428 

.27 

88472 

.44 

31 

30 

77439 

•28 

OQ 

86921 

.44 

A  4 

30 

30 

78445 

.27 

O7 

88498 

.44 

30 

31 

9.77456 

•Zo 

9.86947 

.44 

29 

31 

9.78461 

.Zl 

9.88524 

.44 

29 

32 

77473 

•28 

86974 

.44 

28 

32 

78478 

.27 

88550 

.44 

28 

33 

77490 

•28 

87000 

.44 

27 

33 

78494 

.27 

88577 

.44 

27 

34 

77507 

•28 

87027 

.44 

26 

34 

78510 

.27 

88603 

.44 

26 

35 

77524 

•28 

87053 

.44 

25 

35 

78527 

.27 

88629 

.44 

25 

36 

77541 

•28 

87079 

.44 

24 

36 

78543 

.27 

88655 

.44 

24 

37 

77558 

•  28 

87106 

.44 

23 

37 

78560 

.27 

88681 

.44 

23 

38 

77575  *£! 

87132 

.44 

22 

38 

78576 

.27 

88707 

.44 

22 

39 
40 

77592 
77609 

•28 

87158 
87185 

.44 
.44 

21 
20 

39 
40 

78592 
78609 

.27 
.27 

88733 
88759 

.44 

.44 

21 

20 

41 

42 

9.77626 
77643 

•28 
•28 

9.87211 

87238 

.44 

.44 

19 
18 

41 
42 

9.78625 

78642 

.27 

.27 

9.88786 
88812 

.44 

.44 

19 
18  • 

43 

77660 

.28 

87264 

.44 

17 

43 

78658 

.27 

88838 

.44 

17 

44 

77677 

•28 

87290 

.44 

16 

44 

78674 

.27 
o«7 

88864 

.44 

16 

45 

77694 

•28 

87317 

.44 

15 

45 

•78691 

.Zl 

88890 

.43 

15 

46 
47 

77711 

77728 

•28 
.28 

87343 
87369 

.44 

.44 

14 
13 

46 
47 

78707 

78723 

.27 
.27 

88916 
88942 

.43 
.43 

14 
13 

48 

77744 

•28 

87396 

.44 

12 

48 

78739 

.27 

o- 

88968 

.4:; 

12 

49 

77761 

•  28 

87422 

.44 

11 

49 

78756 

.Zt 
0*7 

88994 

.4.°, 

11 

50 

77778 

.28 

87448 

.44 

10 

50 

78772 

.Zt 

89020 

.4:; 

10 

51 

9.77795 

.28 

9.87475 

.44 

9 

51 

9.78788 

.27 

9.89046 

.43 

9 

52 

77812 

.28 

87501 

.41 

8 

52 

78805 

.27 

89073 

.43 

8 

53 

77829 

.28 

87527 

.44 

7 

53 

78821 

.27 

89099 

.43 

7 

54 

77846 

.28 

87554 

.44 

6 

54 

78837 

.27 

89125 

.43 

6 

55 

77862 

•28 

87580 

.44 

5 

55 

78853 

.27 

89151 

.43 

5 

56 
57 

77879 
77896 

.28 

.28 

87606 
87633 

.44 
.44 

4 
3 

56 
57 

78869 
78886 

.27 
.27 

89177 
89203 

.43 
.43 

4 
3 

58 

77913 

.28 

87659 

.  1  1 

2 

58 

78902 

.27 

89229 

.43 

2 

59 

77930 

.28 

87685 

.44 

1 

59 

78918 

.27 

89255 

.43 

1 

60 

77946 

.28 

87711 

.44 

0 

60 

78934 

.27 

89281 

.43 

0 

M. 

Cosine. 

Cotnng. 

Dl" 

31. 

31. 

Cosine. 

~D1" 

Ootang. 

Dl" 

M. 

53= 


88° 


SINES  AND    TANGENTS. 


30° 


•  At. 

Sine. 

Di" 

Taug. 

Di"   M. 

M.  ; 

Mac. 

Dl" 

Tafag. 

D," 

M. 

0 

1 

9.78934 
78950 

0.27 

9.89281 
89307 

0.43 

6!) 
59 

0 
1 

9*9887 
79903 

0.26 

9.90837 
90S63 

0.43 

60 
59 

2 

78967 

.27 

89333 

.4:1 

58 

2 

79918 

J26 

908S9 

.43 

58 

3 

78983 

.27 

89359 

.43 

57 

3 

79934 

.26 

90914 

.43 

57 

4 

78999 

.27 

89385 

.43 

56 

4 

79950 

.26 

90940 

.43 

56 

5 

79015  ''I 

89411 

.43 

55 

5 

79965 

.26 

90966 

.43 

55 

6 

79031 

.ZY 

89437 

.43 

4>> 

54 

6 

79981 

.26 

cw» 

90992 

.43 

54 

7 

79047 

.27 

89463 

.> 

53 

7 

79996 

.ZO 

91018 

! 

53 

8 

79063 

.27 

89489 

.43 

52 

8 

80012 

.26 

91043 

.43 

52 

9 
10 

79079 
79095 

.27 
.27 

89515 
89541 

.43 

.43 

51 

50 

9 

10 

80027 
80043 

.*26 

91069 
91095 

A3 

51 
50 

11 
12 

9.79111 
79128 

.27 

.27 

07 

9.89567 
89593 

.43 
.43 

49 
48 

11 

12 

9.80058 
80074 

.26 
.26 

9.91121 
91147 

.43 

.43 
.43 

49 

48 

13 
14 

79144 
79160 

.1  i 

.27 

89619 
89645 

.43 

47 
46 

13 
14 

80089 
80105 

.26 

91172 
91198 

'.43 

47 
46 

15 

79176 

•2  i 

89671 

4.-> 

45 

15 

80120 

.zo 

91224 

40 

45 

16 

79192 

.z7 

89697 

0 

44 

16 

80136 

nn 

91250 

*~rG 

44 

17 
18 
19 

20 

79208 
79224 
79240 

79256 

.27 
.27 

.27 
.27 

89723 

89749 
89775 
89801 

.43 
.43 
.43 
.43 

43 
42 
41 
40 

17 
18 
19 
20 

80151 
80166 
80182 
80197 

•  ZU 

.26 
.26 
.26 

91276 
91301 
91327 
913D3 

143 
.43 
.43 

43 
42 
41 
40 

21 

22 

9.79272 

7928S 

.27 
.27 

9.S9S-J7 
89853 

.43 
.43 

39 
38 

21 
22 

9.80213 
80228 

.26 
.26 
.26 

9.91379 
91404 

!43 

.43 

39 

38 

23 

79304 

•2/ 

89879 

.4.5 

37 

23 

80244 

91430 

A  O 

37 

24 
25 

79319 
79335 

.27 
.27 

89905 
89931 

.43 
.43 

36 
35 

24 
25 

80259 
80274 

.26 

91456 
91482 

.4o 
.43 

36 
35 

26 

79351 

.27 

89957 

.43 

34 

26 

80290 

.26 

91507 

.43 

34 

27 

79367 

.27 

89983 

.43 

33 

27 

80305 

.26 

91533 

.43 

A  9 

33 

28 

79383 

.26 

90009 

.43 

32 

28 

80320 

.26 

91559 

.4d 

32 

29 

79399 

.26 

90035 

.43 

31 

29 

80336 

.26 

91585 

.43 

A  O 

31 

30 
31 
32 

79415 
9.79431 

79447 

.26 
.26 
.26 

90061 
9.90086 
90112 

.43 
..43 
.43 

30 

29 
28 

30 
31 
32 

80351 
9.80366 
80382 

.26 
.26 

.26 

91610 
9.91636 
91662 

.43 
.43 
.43 

A  O 

30 
29 

28 

33 
34 
35 
36 
37 
38 
39 
40 
41 
42 

79463 
79478 
79494 
79510 
79526 
79542 
79558 
79573 
9.79589 
79605 

.26 
.26 
.26 
.26 
.26 
.26 
.26 
.26 
.26 
.26 

90138 
90164 
90190 
90216 
90242 
90268 
90294 
90320 
9.90346 
90371 

.43 
.43 
.43 
.43 
.43 
.43 
.43 
.43 
.43 
.43 

27 
26 
25 
24 
23 
22 
21 
20 
19 
18 

33 
34' 
35 
36 
37 
38 
39 
40 
41 
42 

80397 
80412 
80428 
80443 
80458 
80473 
80489 
80504 
9.80519 
80534 

.25 
.25 
.25 
.25 
.25 
.25 
.25 
.25 
.25 
.25 

91683 
91713 
91739 
91765 
91791 
91816 
91842 
91868 
9.91893 
91919 

A6 
.43 
.43 
.43 
.43 
.43 
.43 
.43 
.43 
.43 

27 
26 
25 
24 
23 
22 
21 
20 
19 
18 

43 
44 
45 

79621 
79636 
79652 

.26 
.26 
.26 

90397 
90423 
90449 

.43 
.43 
.43 

17 
16 
15 

43 
44 
45 

80550 
80565 
80580 

•  25 
.25 
.25 

(1C 

91945 
91971 
91996 

.43 
.43 

17 
16 
15 

46 

47 

48 

79668 
79684 
79699 

•26 
.26 
.26 

90475 
90501 
90527 

.43 
.43 
.43 

14 
13 
12 

46 
47 

48 

80595 
80610 
80625 

•  /O 

.25 
.25 

92022 
92048 
92073 

!43 
.43 

14 

13 
12 

49 
50 
51 
52 
53 
54 

79715 
79731 

9.79746 
79762 
79778 
79793 

•26 
.26 
.26 
.26 
.26 
.26 

90553 
90578 
9.90604 
90630 
90656 
90682 

'.43 
.43 
.43 
.43 
.43 

A  O 

11 
10 

9 

8 
7 
6 

49 
50 
51 
52 
53 
54 

80641 
80656 
9.80671 
80686 
80701 
80716 

.25 
.25 

.25 
.25 

.25 

92099 
92125 
9.92150 
92176 
92202 
92227 

!43 
.43 
.43 
.43 
.43 

11 
10 
9 
8 

7 
6 

55 
56 

79809 
79825 

!26 

90708 
90734 

.TEG 

.43 

5 
4 

55 
56 

80731 
80746 

.25 

92253 
92279 

!43 

5 
4 

57 

79840 

o/> 

90759 

A  9 

3 

57 

80762 

OR 

92304 

A  Q 

3 

58 
59 

79856 
79872 

.zo 
.26 

90785 
90811 

Ao 

.43 

40 

2 
1 

58 

59 

80777 
80792 

0£0 

.25 

92330 
92356 

•4o 

.43 

2 
1 

60 

79887 

' 

90837 

G 

0 

60 

80807 

92381 

' 

0 

M. 

Cosine. 

Dl" 

Cotang. 

Dl" 

M. 

M. 

Cosine. 

DI" 

Cotanj*. 

Dl" 

M. 

47 


5O° 


40° 


TABLE  IV.— LOGARITHMIC 


M. 

Situ,. 

Dl" 

Tang. 

Dl" 

M. 

M. 

Sine.   Dl" 

Tang. 

Dl" 

M. 

0 
1 

9.80807 

80822 

0.25 

9.92381 
92407 

O.f3 

60 

59 

0 
1 

9.8  1694 
81709 

G.24 

9.93916 
93942 

0.4.3 

60 

59 

2 

80837 

.25 

92433 

.43 

58 

2 

81723 

.24 

93967 

.43 

58 

3 

80852 

.25 

92458 

.43 

57 

3 

81738 

.24 

93993 

.43 

57 

4 

80867 

.zo 

92484 

.43 

56 

4 

81752 

.24 

94018 

.43 

56 

5 

80882 

.25 

92510 

.43 

55 

5 

81767 

.24 

94044 

.43 

55 

6 

80897 

.25 

92535 

.43 

54 

6 

81781 

.24 

94069 

.43 

54 

7 

80912 

.25 

92561 

.43 

53 

7 

81796 

.24 

94095 

.43 

53 

8 

80927 

.25 

92587 

.43 

52 

8 

81810 

.24 

94120 

.42 

52 

9 
10 

80942 
80957 

.25 
.25 

92612 
•92638 

.43 
.43 

51 
50 

9 
10 

81825 
81839 

.24 
.24 

94146 
94171 

•J2  51 

U  50 

11 

9.80972 

.25 

9.92663 

.43 

49 

11 

9.81854 

.24 

9.94197 

.42 

49 

12 

80987 

.25 

92689 

.43 

48 

12 

81868 

.24 

94222 

.42 

48 

13 

81002 

.25 

92715 

.43 

47 

13 

81882 

.24 

94248 

.42 

47 

14 

81017 

.25 

92740 

.43 

46 

14 

81897 

.24 

94273 

.42 

46 

15 

81032 

.25 

92766 

.43 

45 

15 

81911 

.24 

94299 

.42 

45 

16 

81047 

.25 

92792 

.43 

44 

16 

81926 

.24 

94324 

.42 

44 

17 

81061 

.25 

OK 

92817 

.43 

1  O 

43 

17 

81940 

.24 

94350 

.42 

43 

18 

81076 

ftJEO 

92843 

.43 

42 

18 

81955 

.Z4 

94375 

.42 

42 

19 

81091 

.25 

92868 

.43 

41 

19 

81969 

.24 

94401 

.42 

41 

20 

81106 

.25 

92894 

.43 

40 

20 

81983 

.24 

94426 

.42 

40 

21 

9.81121 

.25 

9.92920 

.43 

39 

21 

9.81998 

.24 

9.94452 

.42 

39 

22 
23 

81136 
81151 

.25 
.25 

92945 
92971 

.43 
.43 

38 
37 

22 
23 

82012 
82026 

Hi  94477 

•fj.   94503 

.42 
.42 

38 
37 

24 
25 

81166 
81180 

.25 
.25 

92996 
93022 

.43 
.43 

36 
35 

24 
25 

82041 
82055 

.24 
.24 

94528 
94554 

.42  '• 

1?  S 

26 

81195 

.25 

93048 

.43 

34 

26 

82069 

.24 

94579 

'42  34 

27 

81210 

.25 

93073 

.43 

33 

27 

82084 

.24 

94604 

An   33 

28 

81225 

.25 

93099 

.43 

32 

28 

82098 

.24 

94630 

.42 

32 

29 

81240 

.25 

93124 

.43 

31 

29 

82112 

.24 

94655 

.42 

31 

30 
31 
32 

81254 
9.81269 
81284 

.25 

.25 
.25 

93150 
9.93175 
93201 

.43 
.43 
.43 

30 

29 

28 

30 
31 

32 

82126 
9.82141 
82155 

.24 
.24 
.24 

94681 
9.94706 
94732 

.42 
.42 

.42 

30 
29 

28 

33 

81299 

.25 

93227 

.43 

27 

33 

82169 

.24 

94757 

.42 

27 

34 

81314 

.25 

93252 

.43 

26 

34 

82184 

.24 

94783 

.42 

26 

35 

81328 

.25 

93278 

.43 

25 

35 

82198 

.24 

94808 

.42 

25 

36 

81343 

.25 

93303 

.43 

24 

36 

82212 

.24 

94834 

.42 

24 

37 

81358 

.25 

93329 

.43 

23 

37 

82226 

.24 

94859 

.42 

23 

38 

81372 

.25 

93354 

.43 

22 

38 

82240 

.Z4 

94884 

.42 

22 

39 

81387 

.25 

93380 

.43 

21 

39 

82255 

.24 

94910 

.42 

21 

40 

81402 

.25 

93406 

.43 

20 

40 

82269 

.24 

94935 

.42 

20 

41 

9.81417 

.25 

9.93431 

.43 

19 

41 

9.82283 

.24 

9.94961 

.42 

19 

42 

81431 

.24 

93457 

.43 

18 

42 

82297 

.24 

94986 

.42 

18 

43 

81446 

.24 

93482 

.43 

17 

43 

82311 

.24 

95012 

.42 

17 

44 

81461 

.24 

93508 

.43 

16 

44 

82326 

.24 

95037 

.42 

16 

45 

81475 

.24 

93533 

.43 

15 

45 

82340 

.z4 

95062 

.42 

15 

46 

81490 

.24 

93559 

.43 

14 

46 

82354 

.24 

95088 

.42 

14 

47 

81505 

.24 

93584 

.43 

13 

47 

82368 

.24 

95113 

.42 

13 

48 

81519 

.24 

93610 

.43 

12 

48 

82382 

.24 

95139 

.42 

12 

49 

81534 

.24 

93636 

.43 

11 

49 

82396 

.24 

95164 

.42 

11 

50 

81549 

.24 

93661 

.43 

10 

50 

82410 

.24 

95190 

.42 

10 

51 

9.81563 

.24 

9.93687 

.43 

9 

51 

9.82424 

.24 

9.95215 

.42 

9 

52 

81578 

.24 

93712 

.43 

8 

52 

82439 

.23 

95240 

.42 

8 

53 

81592 

.24 

93738 

.43 

7 

53 

82453 

.23 

95266 

.42 

7 

54 

81607 

.24 

93763 

.43 

6 

54 

82467 

.23 

95291 

.42 

6 

55 

81622 

.24 

93789 

.43 

5 

55 

82481 

.23 

95317 

.42 

5 

56 

81636 

.24 

93814 

.43 

4 

56 

82495 

.23 

95342 

.42 

4 

57 

58 

81651 
81665 

.24 
.24 

93840 
93865 

.43 
.43 

3 
2 

57 

58 

82509 
82523 

.23 
.23 

95368  '« 
95393'  ™ 

3 
2 

59 

81680 

.24 

93891 

.43 

1 

59 

82537 

.23 

95418 

.1Z 

1 

60 

81694 

.24 

93916 

.43 

0 

60 

82551 

.23 

95444 

.42 

0 

M. 

Cosine. 

D!" 

Ootari!?. 

Dl" 

M. 

M. 

Cosine. 

Dl" 

Cotang. 

Dl" 

M. 

49° 


48 


48* 


SINES  AND  TANGENTS. 


43 


M. 

Sine. 

Dl" 

Tang.  :  Dl" 

M. 

M.   Sine. 

Dl" 

Tang. 

Dl" 

M. 

0 
1 

2 

9.82551 
82565 
82579 

0.23 
.23 

9.95444 
65469 
95495 

0.42 

60 
59 

58 

0 

2 

9.83378 
83392 
83405 

0.23 
.23 

9.96966 
96991 
97016 

0.42 
.42 

60 
59 

58 

3 

82593 

.23 

or) 

95520 

A  O 

57 

3 

83419 

.23 
oo 

97042 

.42 

(  n 

57 

4 

82607 

.ZO 

95545 

.4z 

56 

4 

83432 

.zo 

97067 

.4z 

56 

5 

82621 

.23 

oo 

95571 

.42 

55 

5 

83446 

.23 
oo 

97092 

.42 

A  O 

55 

6 

7 

82635 
82649 

•  Zo 

.23 

95596 
95622 

.42 

54 
53 

6 

7 

83459 
83473 

•  ZO 

.22 

97118 
97143 

AZ 

.42 

54 
53 

8 

82663 

.23 
oo 

95647 

.42 

A  O 

52 

8 

83486 

.22 
oo 

97168 

.42 

A  O 

52 

9 

82677 

.zo 

95672 

.4z 

51 

9 

83500 

.zz 

97193 

.4z 

51 

10 

82691 

.23 

95698 

.42 

A  O 

50 

10 

83513 

.22 

OO 

97219 

.42 

A  O 

50 

11 
12 

9.82705 
82719 

.23 

9.95723 

95748 

AZ 

.42 

49 

48 

11 
12 

9.83527 
83540 

.ZZ 

.22 

9.97244 
97269 

AZ 
.42 

49 

48 

13 
14 

82733 
82747 

.23 
.23 

95774 
95799 

.42 
.42 

47 
46 

13 
14 

83554 
83567 

.22 
.22 

97295 
97320 

.42 
.42 

47 

46 

15 
16 

82761 

82775 

.23 
.23 

95825 
95850 

.42 
.42 

45 

44 

15 
16 

83581 
83594 

.22 
.22 

97345 
97371 

.42 
.42 

45 
44 

17 

82788 

.23 

95875 

.42 

43 

17 

83608 

.22 

97396 

.42 

43 

18 

82802 

.23 

95901 

.42 

42 

18 

83621 

.22 

97421 

.42 

42 

19 

82816 

.23 

95926 

.42 

41 

19 

83634 

.22 

97447 

.42 

41 

20 

82830 

.23 

95952 

.42 

A  O 

40 

20 

83648 

.22 

97472 

.42 

A  O 

40 

21 

9.82844 

.23 

9.95977 

•4z 

39 

21 

9.83661 

.zz 

9.97497 

.4z 

39 

22 

82858 

.23 

96002 

.42 

38 

22 

83674 

.22 

97523 

.42 

38 

23 

82872 

.23 

960281  A* 

37 

23 

83688 

.22 

97548 

.42 

37 

24 

82885 

.23 

96053  1  •;; 

36 

24 

83701 

.22 

97573 

.42 

36 

25 

26 

82899 
82913 

.23 
.23 

96078  « 
96104 

35 
34 

25 
26 

83715 
83728 

.22 
.22 

97598 
97624 

.42 
.42 

35 
34 

27 

28 

82927 
82941 

.23 
.23 

96129 
96155 

.42 

.42 

33 
32 

27 

28 

83741 
83755 

.22 
.22 

97649 
97674 

.42 
.42 

33 
32 

29 

82955 

.23 

96180 

•42 

31 

29 

83768 

.22 

97700 

.42 

31 

30 

82968 

.23 

96205 

.42 

30 

30 

83781 

.22 

97725 

.42 

30 

31 

9.82982 

.23 

9.96231 

•42 

29 

31 

9.83795 

.22 

9.97750 

.42 

29 

32 

82996 

.23 
oo 

96256 

•42 

28 

32 

83808 

.22 

oo 

97776 

.42 

A  O 

28 

33 

83010 

•  zo 

96281 

.42 

27 

33 

83821 

.zz 

97801 

.4z 

27 

34 

83023 

.23 

96307 

.42 

26 

34 

83834 

.22 

97826 

.42 

26 

35 

83037 

.23 

O'-J 

96332 

.42 

25 

35 

83848 

.22 
oo 

97851 

.42 

A  O 

25 

36 

83051 

•  zo 

96357 

.42 

24 

36 

83861 

•  zz 

97877 

.4z 

24 

37 

83065 

•23 

96383 

.42 

23 

37 

83874 

.22 

97902 

.42 

23 

38 

83078 

•23 

96408 

.42 

22 

38 

83887 

•22 

97927 

.42 

22 

39 

83092 

•  23 

96433 

.42 

21 

39 

83901 

•22 

97953 

.42 

21 

40 

83106 

.23 

96459 

.42 

20 

40 

83914 

.22 

97978 

.42 

20 

41 

9.83120 

•23 

9.96484 

.42 

19 

41 

9.83927 

.22 

9.98003 

.42 

19 

42 

43 

83133 
83147 

•23 
.23 

96510 
96535 

.42 

.42 

18 
17 

42 
43 

83940 
83954 

.22 
.22 

98029 
98054 

.42 
.42 

18 
17 

44 

83161 

•  23 

96560 

.42 

16 

44 

83967 

•  22 

98079 

•42 

16 

45 

83174 

•  23 

96586 

.42 

15 

45 

83980 

•22 

98104 

.42 

15 

46 

83188 

•  23 

96611 

.42 

14 

46 

83993 

•  22 

98130 

•42 

14 

47 

83202 

•  23 

96636 

.42 

13 

47 

84006 

.22 

98155 

.42 

13 

48 

83215 

•23 
oo 

96662 

.42 

12 

48 

84020 

•  22 

00 

98180 

.42 

49 

12 

49 

83229 

•  zo 

96687 

•4z 

11 

49 

84033 

•  ZZ 

98206 

•4Z 

11 

50 

83242 

•  23 

96712 

.42 

10 

50 

84046 

•22 

98231 

.42 

10 

51 

9.83256 

•  23 

9.96738 

.42 

9 

51 

9.84059 

•22 

9.98256 

•42 

9 

52 

83270 

•23 

96763 

.42 

8 

52 

84072 

•  22 

98281 

.42 

8 

53 

8328:'. 

•23 

oo 

96788 

.42 

7 

53 

84085 

•  22 

98307 

.42 

7 

54 

83297 

•  Zo 

96814 

•42 

6 

54 

84098 

.22 

98332 

.42 

6 

55 

83310 

•  23 

96839 

.42 

5 

55 

84112 

•  22 

98357 

•42 

5 

56 

83324 

.23 

96864 

.42 

4 

56 

84125 

.22 

98383 

.42 

4 

57 

83338 

.23 

96890 

.42 

3 

57 

84138 

.22 

98408 

.42 

3 

58 

83351 

•  23 

96915 

.42 

2 

58 

84151 

•  22 

98433 

.42 

2 

59 

83365 

.23 

96940 

.42 

1 

59 

84164 

.22 

98458 

.42 

1 

60 

83378 

.23 

96966 

.42 

0 

60 

84177 

.22 

98484 

.42 

0 

M. 

Cosine. 

Dl" 

Cotang. 

"DP 

M. 

M. 

C/osine. 

Dl" 

OotRllg. 

Dl" 

M. 

47C 


49 


46° 


44° 


TABLE  IV.— LOGARITHMIC 


45° 


M. 

Sine.   Dl" 

Tane.   Dl" 

M. 

M. 

Hue. 

Dl"   Tan-. 

Dl"   M. 

0 
1 

9.84177 
84190 

0.22 

9.98484 
9»509 

0.42 

60 
59 

0 
1 

9.84949 
84961 

0.21 

10.00000 
00025 

0.42 

60 
59 

2 

84203 

.22 

98534 

.42 

58 

2 

84974 

.21 

00051 

.42 

58 

3 

84216 

.22 

98560 

.42 

57 

3 

84986 

.21 

06078 

.42 

57 

4 

84229 

.22 

98585 

.42 

56 

4 

84999 

.21 

00101 

.42 

56 

5 

84242 

.22 

98610 

.42 

55 

5 

85012 

.21 

00126 

.42 

55 

6 

84255 

.22 

98635 

.42 

54 

6 

85024 

.21 

00152 

.42 

54 

7 

84269 

.22 

98661 

.42 

53 

7 

85037 

.21 

00177 

.42 

53 

8 

84282 

.22 

98686 

.42 

52 

8 

85049 

.21 

00202 

.42 

52 

9 

84295 

.22 

98711 

.42 

51 

9 

85062 

.21 

00227 

.42 

51 

10 

84308 

.22 

98737 

.42 

50 

10 

85074 

.21 

00253 

.42 

50 

11 

9.84321 

.22 

.1.) 

9.98762 

.42 

«  o 

49 

11 

9.85087 

.21 
01 

10.00278 

.42 

49 

12 

84334 

.zz 

98787 

«4z 

48 

12 

85100 

.zl 

00303 

.42 

48 

13 
14 
15 

84347 
84360 
84373 

.22 
.22 
.22 

98812 
98838 
98863 

.42 
.42 
.42 

47 
46 
45 

13 
14 

15 

85112 
85125 
85137 

.21 
.21 
.21 

00328  •** 
00354  •** 
00379  •** 

47 
46 
45 

16 

84385 

.22 

98888 

.42 

44 

16 

85150 

00404  '-J; 

44 

17 

84398 

.22 

93913 

.42 

43 

17 

85162 

00430  •** 

43 

18 

84411 

.22 

98939 

.42 

42 

18 

85175 

•;}    00455  '**!  42 

19 

84424 

.22 

98964 

.42 

41 

19 

85187 

00480 

.42 

41 

20 

S4437 

.zz 

98989 

.42 

40 

20 

85200  '*\  \   00505 

.42 

40 

21 

9.84450 

.22 

99 

9.99015 

.42 

39 

21 

9.852121  '^|  10.  00531 

.42 

39 

22 

84463 

.ZZ 

99040 

Ai 

38 

22 

85225  •*  1   00556 

.4z 

38 

23 

84476 

.22 

99065 

.42 

37 

23 

85237  •£}•   00581 

.42 

37 

24 

84489 

.21 
01 

99090 

.42 

A  O 

36 

24 

85250  'i\  \   00606 

.42 

36 

25 

84502 

.zl 

99116 

.4z 

35 

25 

85262 

00632 

.42 

35 

26 

84515 

.21 

99141 

.42 

34 

26 

85274  "i\\   00657  '^ 

34 

27 
28 
29 
30 

84528 
84540 
84553 
84566 

.21 
.21 
.21 
.21 

99166 
99191 
99217 
99242 

.42 
.42 
.42 
.42 

33 
32 
31 
30 

27 
28 
29 
30 

85287 
85299 
85312 
85324 

5    006S2  '- 

1  1   °°707  "42 

1    °°7:"  *42 
•J'l   007581  -JJ 

33 
32 
31 
30 

31 

9.84579 

.21 

9.99267 

.42 

29 

31 

9.85337 

•i\:  10.00783  '?f  29 

32 

84592 

.21 

O  1 

99293 

.42 

1  O 

28 

32 

85349  •£!    00809J  •**  28 

33 

84605 

.2  1 

01 

99318 

.4z 

27 

33 

8536  IS  i{    00834 

27 

34 

84618 

.zl 

99343 

.42 

26 

34 

85374  •*!    00859  '^!  26 

35 

84630 

.21 

99368 

.42 

25 

35 

85386 

•*\    00884  •**  25 

36 

84643 

.21 

99394 

.42 

24 

36 

85399 

•;    00910  ™  24 

37 
38 
39 

84656 
84669 
84682 

.21 
.21 
.21 

99419 
99444 
99469 

.42 

.42 
.42 

23 
22 
21 

37 
38 
39 

854  1  1 
85423 
85436 

.21 
.21 

1! 

00935  *JJ 
00960  •« 
00985  ™J 

23 
22 
21 

40 

84694 

.21 

99195 

.42 

20 

40 

85448 

.zl 

01011  I  •*: 

20 

41 

9.84707 

.21 

9.99520 

.42 

19 

41 

9.85460 

•r,!  10.01036!  -q.l 

19 

42 

84720 

.21 

99545 

.42 

18 

42 

85473 

I   °1061  1? 

18 

43 

84733 

.21 

99570 

.42 

17 

43 

85485 

•\\  olosr  •?; 

17 

44 

84745 

.21 

99596 

.42 

16 

44 

85497 

.21 

Ct  1 

01112 

•** 

16 

45 

84758 

.21 

99621 

.42 

15 

45 

85510  't\    01137 

•42  ,, 

4  0  '    *  " 

46 

84771 

.21 

99646 

.42 

14 

46 

85522 

20   °1162 

.42 

14 

47 

84784 

.21 

99672 

.42 

13 

47 

85534 

OA 

01188 

.42 

13 

48 

84796 

.21 

99697 

.42 

12 

48 

85547 

.20 

01213 

.42 

12 

49 

84809 

.21 

99722 

.42 

11 

49 

85559 

1o    01238 

.42 

11 

50 

84822 

.21 

99747 

.42 

10 

50 

85571 

'on    01263  '  '" 

10 

51 

9.84835 

.21 

9.99773 

.42 

9 

51 

9.85583 

•£!  1  10.  01  289  -JJ 

9 

52 

84847 

.21 

99798 

.42 

8 

52 

85596 

01314   A1 

8 

53 

84860 

.21 

99823 

.42 

7 

53 

85608 

••JJ   111  33ii  -I1; 

7 

54 

84873 

.21 

99848 

.42 

6 

54 

85620 

•|J    01365'  •  r, 

6 

55 

84885 

.21 

99874 

.42 

5 

55 

85632 

.Z() 

01390)  "} 

5 

56 

84898 

.21 

99899 

.42 

4 

56 

85645 

.20 

01415!  'f 

4 

57 

84911 

.21 

99924 

.42 

3 

57 

85657 

.20 

01440 

3 

84923 

.21 

99949 

.42 

2 

58 

85669 

.20 

01466  'J* 

2 

59 

84936 

.21 

999^(5 

.42 

59 

85681 

.20 

01  -I'M   'JJ 

1 

60 

84949 

.21 

10.00000 

.42 

0 

60 

85693 

.20 

01  5  Mi:  ' 

0 

M. 

Cosine. 

Dl" 

Cntnnsc. 

Dl" 

M. 

M. 

Cosino. 

Dl"  '  ('otaim.   I'i" 

M. 

44C 


SINES  AND  TANGENTS. 


47° 


31. 

Sine.    Dl" 

Tang. 

Dl" 

M. 

M. 

Sine. 

Dl" 

Tang. 

Dl" 

M 

0 
1 

9.85693 
85708 

0.20 

10.01516 
01542 

0.42 

,1  O 

60 

59 

0 
i 

9.86413 

86425 

0.20 
on 

10.03034 
03060 

0.42 

A  O 

60 
59 

2 

85718 

.20 

01567 

AL 

58 

2 

86436 

.20 

03085 

AL 

58 

3 

4 

85730 

85742 

.20 
.20 

01592 

01617 

.42 
.42 

57 
56 

3 
4 

86448!  'f 
864601  ^JJ 

03110 
03U6 

.42 

.42 

57 
56 

5 

85754 

.20 

90 

01643 

.42 

4.9 

55 

5 

86472  '2 

03161 

.42 

4.9 

55 

6 

85766 

•  ZU 

01668 

AL 

54 

6 

86483 

03186 

AL 

54 

7 

85779 

.20 

90 

01693 

.42 

53 

7 

86495 

.20 
on 

03212 

.42 

4.9 

53 

8 

85791 

.zu 

01719 

AL 

52 

8 

86507  '*" 

03237 

AL 

52 

9 

85803 

.20 

01744 

.42 

51 

9 

86518  -;J 

03262 

.42 

51 

10 

85815 

.20 

01769 

.42 

50 

10 

86530   „!! 

03288 

.42 

50 

11 

9.85827 

.20 
.20 

fO.01794 

.42 
49 

49 

11 

9.86542 

.ZU 

on 

10.03313 

.42 

4.9 

49 

12 

85839 

01820 

AL 

48 

12 

86554  '*vn- 

03338 

AL 

48 

13 

85851 

.20 

90 

01845 

J.42 
A  O 

47 

13 

86565 

03364 

.42 

47 

14 

85864 

01870 

AL 

46 

14 

86577 

19 
'in 

03389 

.42 

46 

15 

85876 

.20 

01896 

.42 

A  O 

45 

15 

86589 

.19 

03414 

.42 

45 

16 

85888 

.20 

01921 

AL 

44 

16 

86600 

.19 

03440 

.4- 

44 

17 

85900 

.20 

01946 

.42 

43 

17 

8661.2!  '}j! 

03465 

.42 

43 

18 

85912 

.20 

01971 

.42 

42 

18 

86624!  •:• 

03490 

.42 

42 

19 

85924 

.20 

01997 

.42 

41 

19 

86635 

.jy 

03516 

.42 

41 

20 
21 
22 

85936 
9.85948 
85960 

.20 
.20 
.20 

on 

02022 
10.02047 
02073 

.42 
.42 
.42 

40 
39 
38 

20 
21 

22 

86647 
9.866.59 
86670 

.19 
.19 

i  n 

03541 
10.03567 
03592 

.42 
.42 

40 
39 

38 

23 
24 

85972 
85984 

.2" 
.20 

02098  '** 
02123   .* 

37 
36 

23 

24 

86682 

86694 

.19 

.19 

03617 
03643 

.42 

.42 

37 
36 

25 
26 

85996 
86008 

.20 

02149 
02174 

A-L 
.42 

35 
34 

25 

26 

86705 
86717 

.19 

.19 

03668 
03693 

.42 

35 
34 

27 

86020 

.20 

02199 

.42 

33 

27 

86728 

.19 

03719 

.42 

33 

28 

86032 

.20 

02224 

.42 

32 

28 

86740 

.19 

03744 

.42 

32 

29 

86044 

.20 

02250 

.42 

31 

29 

86752 

'}J 

03769 

.42 

31 

30 

86056 

.20 

90 

02275 

A  O 

30 

30 

86763 

.19 

1  0 

03795 

.42 

A  0 

30 

31 

9.86068 

10.02300 

AL 

29 

31 

9.86775 

•  :r.  10.03820 

AL 

29 

32 

33 

86080 
86092 

.20 
.20 

02326 
02351 

.42 
.42 

28 
27 

32 
33 

86786 
86798 

03845 
03871 

.42 
.42 

28 
27 

34 

86104 

.20 

02376 

.42 

26 

34 

86809 

.19 

03896 

.42 

26 

35 

S61  1(1 

.20 
90 

02402 

.42 

A  O 

25 

35 

86821 

.19 

-i  o 

03922 

.42 

A  O 

25 

36 

'  86128 

02427 

AL 

24 

36 

86832 

.1  .' 

03947 

AL 

24 

87 

86140 

.20 

02452 

.42 

23 

37 

86844 

.19 

03972 

.42 

23 

38 
39 

86152 

86164 

.20 
.20 

on 

02477 
02503 

.42 
.42 

22 
21 

38 
39 

86855 

86867 

.19 
.19 

03998 
04023 

.42 

.42 

22 
21 

40 

86176 

.20 

02528 

.42 

20 

40 

86879 

04048 

.42 

20 

41 

9.86188 

.20 

10.02553 

.42 

19 

41 

9.86890 

'Jo!  10.04074 

.42 

19 

42 

86200 

.20 

on 

02579 

.42 

18 

42 

86902 

.19 

04099 

.42 

18 

43 

86211 

.zU 
on 

02604 

.42 

17 

43 

86913 

.19 

04125 

.42 

17 

44 

86223 

.20 

'  02629 

.42 

16 

44 

86924 

.19 

04150 

.42 

16 

45 

86235 

.20 

02655 

.42 

15 

45 

86936 

.19 

04175 

•42 

15 

46 

86247 

.20 

02680 

.42 

14 

46 

86947 

.19 

04201 

.42 

14 

47 

48 

86259 
86271 

.20 
.20 

OA 

02705 
02731 

.42 
.42 

13 
12 

47 

48 

86959 
86970 

.19 
.19 

04226 
04252 

.42 
.42 

13 
12 

49 

50 

86283 
86295 

.ZO 

.20 

02756 
02781 

.42 
.42 

11 
10 

49 
50 

86982 
86993 

.19 
.19 

04^7 
04302 

.42 

.42 

11 
10 

51 

9.  863  06 

.20 

10.02807 

.42 

9 

51 

9.87005 

.19 

10.04328 

.42 

9 

52 

86318 

OA 

02832 

.42 

8 

52 

87016 

.19 

04353 

.42 

8 

53 

86330 

.20 

02857 

.42 

7 

53 

87028 

.19 

04378 

•42 

7 

54 

86342 

.20 

02882 

.42 

6 

54 

87039 

.19 

04404 

.42 

.  6 

55 

86354 

.20 

02908 

.42 

5 

55 

87050 

.19 

04429 

•42 

5 

56 

86366 

.20 

02933 

.42 

4 

56 

87062 

.19 

04455 

.42 

4 

57 

86377 

.20 

02958 

.42 

3 

57 

87073 

.19 

04480 

.42 

3 

58 

86389 

.20 

02984 

.42 

2 

58 

87085 

.19 

04505 

.42 

2 

59 

86401 

.20 

03009 

.42 

1 

59 

87096 

.19 

04531 

.42 

1 

60 

86413 

.20 

03034 

.42 

0 

60 

87107 

.19 

04556 

.42 

0 

M.   Cosine. 

Dl" 

Cotang. 

Dl" 

31. 

M. 

Cosi7ie. 

Dl"   Cotang. 

Dl" 

M. 

43° 


42° 


TABLE  IV.— LOGARITHMIC 


M. 

Sino. 

l)i" 

Tan?. 

PI" 

M. 

M. 

Sine. 

1)1" 

Tang. 

1)1" 

It. 

0 
1 

9.87107 
87119 

0.19 

10.0455(3 
04582 

0.42 

60 
59 

0 
1 

9.87778 
87789 

0.18 

10.06084 
06109 

0.43 

60 
59 

2 

87130 

.19 

04607 

'42  58 

2 

87800 

.18 

06135 

.43 

58 

3 
4 

87141 
87153 

.19 
.19 

04632 
04658 

^42 

57 
56 

3 

4 

87811 

87822 

.18 

.18 

1  Q 

06160 
06186 

.43 
.43 

A  o 

57 
56 

5 

87164 

.19 

04683 

.42 

55 

5 

87833 

.  1  0 

06211 

.4o 

55 

6 

87175 

.19 

04709 

.42 

54 

6 

87844 

.18 

1  o 

06237 

.43 
40 

54 

7 
8 

87187 
87198 

!l9 

04734 
04760 

!42 

53 
52 

7 
8 

87855 
87866 

•  lo 

.18 

06262 

06288 

.40 
.43 

53 

52 

9 

87209 

19 

04785 

*42 

51 

9 

87877 

.18 

I  O 

06313 

.43 

AQ 

51 

10 
11 

87221 
9.87232 

J9 
1Q 

04810 
10.04836 

!42 

50 
49 

10 
11 

87887 
9.87898 

.  1  o 

.18 

1  o 

06339 
10.06364 

••TO 

.43 

A<\ 

50 
49 

12 
13 
14 

87243 
87255 
87266 

.  1  «7 

.19 
.19 
.19 

04861 
04887 
04912 

^42 
.42 

48 
47 
46 

12 
13 

14 

87909 
87920 
87931 

.  lo 

.18 

.18 

I  Q 

06390 
06416 
06441 

•4o 

.43 
.43 

48 
47 
46 

15 
16 
17 
18 
19 

87277 
87288 
87300 
87311 
87322 

!l9 
.19 
.19 
.19 
1  Q 

04938 
04963 
04988 
05014 
05039 

.42 
.42 
.42 
.42 

45 
44 
43 
42 
41 

15 
16 
17 
18 
19 

87942 
87953 
87964 
87975 
87985 

.Jo 

.18 
.18 
.18 
.18 
1  8 

06467 
06492 
06518 
06543 
06569 

!43 
.43 
.43 
.43 

45 
44 
43 
42 
41 

20 
21 

87334 
9.87345 

.1*7 

.19 

05065 
10.05090 

'.42 

40 
39 

20 
21 

87996 
9.88007 

.1  o 

.18 

06594 
10.06620 

'.43 

40 
39 

22 

87356 

.19 
.19 

05116 

.42 

38 

22 

88018 

.18 

1  u 

06646 

.43 

38 

23 

87367 

05141 

40 

37 

23 

88029 

•  1O 

1  ft 

06671 

A'\ 

37 

24 

87378 

1Q 

05166 

- 

36 

24 

88040 

.  1  O 
1  o 

.  06697 

.4o 

36 

25 

87390 

.19 

05192 

49 

35 

25 

88051 

•  1  O 
I  Q 

06722 

A*t 

35 

26 

27 

87401 
87412 

!l9 
1  Q 

05217 
05243 

!42 

34 

33 

26 
27 

88061 
88072 

•  1O 

.18 

I  Q 

06748 
06773 

.4*-* 

.43 

4°, 

34 
33 

28 

87423 

•  i  y 
.19 

05268 

*49 

32 

28 

88083 

.  10 

1  o 

06799 

A'J 

32 

29 

87434 

05294 

•** 

31 

29 

88094 

•  1  o 

06825 

.4o 

31 

30 

87446 

.19 

05319 

.42 

30 

30 

88105 

•18 

1  Q 

06850 

.43 

30 

31 
32 
33 

9.87457 
87468 
87479 

.19 
.19 
.19 
1  Q 

10.05345 
05370 
05396 

.42 
.42 
.42 

JO 

29 
28 
27 

31 
32 
33 

9.88115 
88126 
88137 

.lo 

.18 
.18 
18 

10.06876 
06901 
06927 

.43 

.43 
.43 

29 

28 
27 

34 
35 
36 

87490 
87501 
87513 

•  i  y 

.19 
.19 
1  Q 

05421 
05446 
05472 

•42 

.42 
.42 

49 

26 
25 
24 

34 
35 
36 

88148 
88158 
88169 

•  lo 
.18 
.18 
.18 

06952 
06978 
07004 

!43 
.43 

26 
25 
24  ' 

37 

38 

87524 
87535 

•  i  y 
.19 

1  Q 

05497 
05523 

•~LZ 

.42 

23 
22 

37 

38 

88180 
88191 

.18 

1  S 

07029 
07055 

!43 

23 
22 

39 
40 
41 

87546 
87557 

9.87568 

-  iy 
.19 
.19 
i  q 

05548 
05574 
10.05599 

.42 
.42 

21 
20 
19 

39 
40 
41 

88201 
88212 
9.88223 

•  1  o 

.18 
•18 
18 

07080 
07106 
10.07132 

!43 
.43 

1H 

21 
20 
19 

42 

87579 

•  iy 

05625 

Ati 

18 

42 

88234 

.10 

07157 

.•iO 

18 

43 

87590 

.18 

05650 

.42 

17 

43 

88244 

.18 

1  Q 

07183 

.43 

17 

44 

87601 

.18 

1  Q 

05676 

'.42 

A  O 

16 

44 

88255 

.lo 

•I  0 

07208 

.43 

A  *J 

16 

45 

46 
47 

87613 
87624 
87635 

.  lo 
.18 
.18 

1  Q 

05701 
05727 
05752 

•42 

.42 
.42 

4  O 

15 
14 
13 

45 
46 
47 

88266 
88276 
88287 

.10 

.18 
.18 

1  o 

07234 
07260 
07285 

.4o 

.43 
.43 

A  O. 

15 
14 
13 

48 

87646 

.lo 

05778 

Ai 

12 

48 

88298 

.lo 

-<  rt 

07311 

A6 

12 

49 

876^- 

.18 

05803 

.42 

11 

49 

883081  '!° 

07337 

.43 

11 

50 

S7668 

.18 
1  8 

05829 

.42 

10 

50 

88319!  'JJ 

07362 

.43 

40 

10 

51 

9.87679 

.  10 

1  Q 

10.05854 

49 

9 

51 

9.88330 

1  o 

10.07388 

43   9 

52 

87690 

•  lo 

1  Q 

05880 

. 

8 

52 

88340 

*1O 

-I  Q 

07413 

A  O 

8 

53 

87701 

.13 

1  ft 

05905 

•™ 

7 

53 

88351 

•  lo 
18. 

07439 

A6 

7 

54 

87712 

•  lo 

05931 

.4o 

6 

54 

88362 

•  10 

07465 

.4o 

6 

55 

87723 

.18 
i  ft 

05956 

.43 

JO 

5 

55 

88372 

.18 

1Q 

07490 

.43 

5 

56 

87734 

.10 

05982 

.4o 

4 

56 

88383  '!" 

07516 

•** 

4 

57 

87745 

.18 

06007 

.43 

3 

57 

88394  *!Q 

07542 

.43 

3 

58 

87756 

.18 

06033 

.43 

2 

58 

88404  'I* 

07567 

.43 

2 

59 

87767 

.18 

06058 

.43 

1 

59 

88415  '\l 

07593 

.43 

1 

60 

87778 

.18 

OfiO«4 

.43 

0 

60 

88425  i  ' 

07619 

.43 

0 

M. 

Cosine. 

Dl" 

CotniiS. 

PI" 

M. 

M. 

Cosine.  PI" 

<"otanc.   PI" 

M. 

41C 


52 


50° 


SINES  AND    TANGENTS. 


M 

bine. 

Dl" 

TaiiK.    Dl"  |  M. 

M. 

Sine. 

Dl" 

Tang. 

D!" 

M. 

0 
1 

9.88425 
88436 

0.18 

10.07619 
07644 

0.43 

60 
59 

0 

1 

9.89050 
89060 

0.17 

10.09163 
09189 

0.43 

60 
59 

2 

88447 

.18 

07670 

.43 

58 

2 

89071 

.17 

09215 

.48 

58 

3 

88457 

.18 

07696 

.43 

57 

3 

89081 

.17 

09241 

.43 

57 

4 

88468 

.18 

07721 

.43 

56 

4 

89091 

.17 

09266 

.43 

56 

5 

88478 

.18 

07747 

.43 

55 

5 

89101 

.17 

09292 

.43 

55 

6 

88489 

.18 

07773 

.43 

54 

6 

89112 

.17 

09318 

.43 

54 

7 

88499 

.18 

07798 

.43 

53 

7 

89122 

.17 

09344 

.43 

53 

8 

88510 

.18 

07824 

.43 

52 

8 

89132 

.17 

09370 

.43 

52 

9 

88521 

.18 

07850 

.43 

51 

9 

89142 

.17 

09396 

.43 

51 

10 

88531 

.18 

07875 

.43 

50 

10 

89152 

.17 

09422 

.43 

50 

11 

9.88542 

.18 

10.07901 

.43 

49 

11 

9.89162 

.17 

10.09447 

.43 

49 

12 

88552 

.18 

07927 

.43 

48 

12 

89173 

.17 

09473 

.43 

48 

13 

88563 

.18 

07952 

.43 

47 

13 

89183 

.17 

09499 

.43 

47 

14 

88573 

.18 

07978 

'X 

46 

14 

89193 

.17 

09525 

.43 

46 

15 

88584 

.18 

1  7 

08004 

.43 

45 

15 

89203 

.17 

09551 

.43 

A  O 

45 

16 

88594 

.  1  / 

08029 

.43 

44 

16   89213 

.17 

09577 

.4o 

44 

17 

88605 

.17 

08055 

.43 

43 

17    89223 

.17 

09603 

.43 

43 

18 

88615 

.17 

08081 

.43 

42 

18 

89233 

.17 

09629 

.43 

42 

19 

88626 

.17 

08107 

•s 

41 

19 

89244 

.17 

09654 

.43 

41 

20 

88636 

.17 

08132 

.43 

40 

20 

89254 

.17 

09680 

.43 

40 

21 

9.88647 

.17 

1  7 

10.08158 

.43 

A  «> 

39 

21 

9.89264 

•}l  110.09706 

.43 

A  O 

39 

22 

88657 

.1  1 

08184 

.43 

38 

22 

89274 

.  1  1 

09732 

.4o 

38 

23 

8866* 

.17 

08209 

.43 

37 

23 

89284 

.17 

09758 

.43 

37 

24 

88678 

.17 

08235 

.43 

36 

24 

89294 

.17 

09784 

.43 

36 

25 

88688 

.17 

08261 

.43 

35 

25 

89304 

•  17 

09810 

.43 

35 

26 
27 

88699 
88709 

.17 
.17 

08287 
08312 

.43 
.43 

34 
33 

26 

27 

89314 
89324 

'}l\   09836 

';:  09862 

.43 
.43 

34 
33 

28 

88720 

.17 

08338 

.43 

32 

28 

89334 

.1  i 

09888 

.43 

32 

29 

88730 

.17 

08364 

.43 

31 

29 

89344 

.17 

09914 

.43 

31 

30 

88741 

.17 

08390 

.43 

30 

30 

89354 

.17 

09939 

.43 

30 

31 

9.88751 

.17 

10.08415 

.43 

29 

31 

9.89364 

•17 

10.09965 

.43 

29 

32 

88761 

08441 

.43 

28 

32 

89375 

.17 

09991 

.43 

28 

33 

88772 

.17 

08467 

.43 

27 

33 

89385 

.17 

10017 

.43 

27 

34 

88782 

08493 

.43 

26 

34 

89395 

•  17 

10043 

.43 

26 

35 

88793 

.17 

08518 

.4.3 

25 

35 

89405 

•  17 

10069 

.43 

25 

36 

88803 

.17 

1  7 

08544 

.43 

24 

36 

89415 

.17 

10095 

.43 

24 

37 

88813 

.1  / 

08570 

.43 

23 

37 

89425 

.17 

10121 

.43 

23 

38 

88824 

08596 

.43 

22 

38 

89435 

•17 

10147 

.43 

22 

39 

88834 

.17 

1  7 

08621 

.43 

21 

39 

89445 

•  17 

10173 

.43 

21 

40 

88844 

.17 

Ni 

08647 

.43 

20 

40 

89455 

•17 

10199 

.43 

20 

41 

9.88855 

10.08673 

•s 

19 

41 

9.89465 

•17 

10.10225 

.43 

19 

42 

88865 

•  17 

-1  7 

08699!  '*!! 

18 

42 

89475 

•  17 

10251 

.43 

18 

43 

88875 

.]  / 

1  7 

08724 

.±6 

17 

43 

89485 

•17 

i  - 

10277 

.43 

17 

44 

88886 

.1  I 

08750 

.43 

16 

44 

89495 

•  17 

10303 

.43 

16 

45 

88896 

.17 

1  7 

08776 

.43 

15 

45 

89504 

•  17 

10329 

.43 

15 

46 

88906 

.  I  I 

08802 

.43 

14 

46 

89514 

.17 

10355 

.43 

14 

47 

88917 

.17 

1  7 

08828 

.43 

13 

47 

89524 

.17 

10381 

.43 

13 

48 
49 

88927 
88937 

.!< 
.17 

08853 
08879 

.43 
.43 

12 
11 

48 
49 

89534 
89544 

•  17 
.17 

10407 
10433 

.43 
.43 

12 
11 

50 

88948 

.17 

08905 

.43 

10 

50 

89554 

•  17 

10459 

.43 

10 

51 

9.88958 

17 

10.08931 

.43 

y|O 

9 

51 

9.89564 

•  17 

1  7 

10.10485 

.43 

9 

52 

88968 

•  i  t 

08957 

.43 

8 

52 

89574 

•  LI 

10511 

.43 

8 

53 

88978 

.17 

1^ 

08982 

.43 

7 

53 

89584 

.17 

10537 

.43 

7 

54 

88989 

i 

09008 

.43 

6 

54 

89594 

•  17 

10563 

.43 

6 

55 

88999 

.17 

09034 

.43 

5 

55 

89604 

.17 

10589 

.43 

5 

56 

89009 

.17 

09060 

.43 

4 

56 

89614 

.16 

10615 

.43 

4 

57 

89020 

09086 

.43 

3 

57 

89624 

.16 

10641 

.43 

3 

58 

89030 

.17 

09111 

.43 

2 

58 

89633 

.16 

10667 

.43 

2 

59 

89040 

.17 

1  7 

09137 

.43 

A  0 

1 

59 

89643 

.16 

10693 

.43 

1 

60 

89050 

.17 

09163  /*' 

0 

60 

89653 

.16 

10719 

.43 

0 

M. 

Cosine. 

Dl" 

GotaiiKi  i  Dl" 

M. 

M. 

Cosine. 

Dl" 

Cot;uiK. 

Dl"   M. 

39 


S.  N.  39. 


53 


38 


52° 


TABLE  IV.— LOGARITHMIC 


M. 

Miie. 

pr 

'fa  ti  jr. 

!»,' 

M. 

M.   Muf.   1>."   'lituir.   In"   M. 

0 

9.89653 

10.10719 

60 

0  \  9.90235  L  „  „ 

10.122hH|  ,,60 

1 

89663 

0.16 

10745 

0.43 

59 

1 

90244  ,  }j| 

123151™:  59 

2 

89673 

.16 

10771 

.43 

58 

2 

90254  'JJ 

12341 

44  58 

3 

69683 

.16 

10797 

.43 

57 

3 

90263  •'? 

12367 

57 

4 

89693 

.16 

1  0823 

.43 

56 

4 

902731  4* 

12394 

.44 

56 

5 

89702 

.16 

10849 

.43 

55 

5 

90282 

.10 

12420 

.44 

55 

6 

7 

89712 
89722 

.16 
.16 

10875 
10901 

.43 
.43 

54 
53 

6 

7 

90292 
90301 

.16 

.16 

12446 
12473 

.44 
.44 

54 
53 

8 

89732 

.16 

10927 

.43 

52 

8 

90311  •'? 

12499 

.44 

52 

9 

89742 

.16 

10954 

.43 

51 

9 

90320  '}J 

12525 

.44 

51 

10 

89752 

.16 

10980 

.43 

50 

10 

90330!  '*2 

12552 

.44 

50 

11 

9.89761 

.16 

10.11006 

.43 

49 

11 

9.90339 

.10 

10.12578 

.44 

49 

12 

89771 

.16 

11032 

.43 

48 

12 

90349 

.16 

12604 

.44 

48 

13 

89781 

.16 

11058 

.43 

47 

13 

90356 

.16 

12631 

.44 

47 

14 

89791J  'IJ 

11084 

.43 

A  *J 

46 

14 

90368 

.16 

1  C 

12657 

.44 

4.1 

46 

15 

89801 

•JU 

11110 

.4o 

45 

15 

90377 

•  JO 

.  rt 

1  2683 

.4-4 

45 

16 

89810 

.16 

11136 

.44 

44 

16 

90386 

.10 

12710 

11  44 

17 

89820 

.16 

11162 

.44 

43 

17 

90396 

.16 

12736 

AA  43 

18 

89830 

.16 

11188 

.44 

42 

18 

90405 

.16 
i  /» 

12762 

11  i  42 

J9 

89840 

.16 

11214 

.44 

41 

19 

90415 

.lo 

12789 

'5  41 

20 

89849 

.16 

«  rt 

11241 

.44 

40 

20 

90424 

.16 

12815 

*S  40 

21 

9.89859  1  \J" 

10.11267 

.44 

39 

21 

9.90434 

•]j|  10.12842 

39 

22 

89869 

11293 

.44 

38 

22 

90443 

.  1  0 

12868 

11  38 

23 

89879 

.16 

11319 

.44 

37 

23 

90452 

.16 
i  /» 

12894 

.44  o7 
ii  ** 

24 

89888 

.16 

11345 

.44 

36 

24 

90462 

.]b 

12921 

.44 

36 

25 

89898 

.16 

11371 

.44 

35 

25 

90471 

.16 

12947 

.44 

35 

26 

89908 

.16 

11397 

.44 

34 

26 

90480 

.16 

12973 

.44 

34 

27 

89918 

.16 

11423 

.44 

33 

27 

90490  'JJ 

13000 

.44 

33 

28 

89927 

.16 

11450 

.44 

32 

28 

90499   ,J 

13026 

.44 

32 

29 

89937 

.16 
i  c 

11476 

.44 

31 

29 

90509  j  '{5 

13053 

..44 

A  1 

31 

30 

89947 

Jo 

11502 

.44 

30 

30 

905181  4" 

13079 

.44 

A  4 

30 

31 
32 

0.89956 
89966 

.16 
.16 

10.11528 
11554 

2 

29 

28 

31 
32 

9.90527 
90537 

.10 

.16 

10.13106 
13132 

.44 
.44 

29 

28 

34 

89976 
89985 

.16 
.16 

11580 
11607 

.44 
.44 

27 
26 

33 
34 

90546 
90555 

.16 

.16 
i  f* 

13158 
13185 

.44 

.44 

4  A 

27 
26 

35 

89995 

.16 

1  A 

1  1  633 

.44 

25 

35 

90565 

.1  0 

13211 

.44 

4  1 

25 

36 
37 

90005 
90014 

.lo 
.16 

11659 
11685 

.44 
.44 

24 

23 

36 
37 

90574 
90583 

.16 

1  £ 

1  3238 
1  3264 

.44 
.44 
1  1 

24 
23 

38 
39 

90024 
90034 

.16 
.16 

11711 
11738 

.44 
.44 

22 
21 

38 
39 

90592 
90602 

*  1  0 

.15 

1  £. 

13291 
13317 

.44 
.44 

22 
21 

40 
41 
42 

90043 
9.90053 
90063 

.16 
.16 
.16 

11764 
10.11790 
11816 

.44 
.44 
.44 

20 
19 

18 

40 
41 
42 

90611 
9.90620 
90630 

.10 

.15 
.15 

1  £. 

13344 
10.13370 
13397 

.44 
.44 
.44 

20 
19 
18 

43 

90072 

.16 

11842 

.44 

17 

43 

•  90639 

•  10 

1  *v 

134231  '** 

17 

44 

45 

90082 
90091 

.16 
.16 

11869 
11895 

.44 

.44 

16 
15 

44 
45 

90648 
90657 

•  10 

.15 

1  ^ 

1  3449 
13476 

.44 

16 
15 

46 

90101 

.16 

11921 

.44 

14 

46 

90667 

JO 
1  ^ 

13502 

.44 

11 

14 

47 

48 

90111 
90120 

.16 
.16 

11947 
11973 

.44 
.44 

13 
12 

47 
48 

90676 
90685 

•  10 

.15 

1  £ 

13529 
1  3555 

.44 

.44 

A  A 

13 
12 

49 

90130 

.16 

12000 

.44 

11 

49 

90694 

.10 

1  £ 

13582 

.44 

11 

11 

50 
51 

90139 
9.90149 

.16 
.16 

12026 
10.12052 

.44 
.44 

10 

9 

50 
51 

90704 
9.90713 

.  1  0 

.15 

1  £ 

13608 
10.13635 

.44 

.44 

11 

10 
9 

52 
53 
54 

90159 
90168 
90178 

.16 
.16 
.16 

12078 
12105 
12131 

.44 
.44 
.44 

8 
7 
6 

52 
53 
54 

90722 
90731 
90741 

.10 

.15 
.15 

i  n 

13662 
13688 
13715 

.44 

.44 

.44 

8 
7 
6 

55 
56 

90187 
90197 

.16 
.16 

12157 
12183 

.44 
.44 

5 
4 

55 

56 

90750 
907f>9' 

•  10 

.15 

13741 
13768 

.44 

5 
4 

57 
58 
59 

90206 
90216 
90225 

.16 
.16 
.16 

12210 
12236 
12262 

.44 
.44 
.44 

3 
2 
1 

57 
58 
59 

90768 
90777 
90787 

.15 
.15 
.15 

13794 
13821 
13847 

.44 
.44 

A  A 

3 
2 

1 

60 

90235 

.16 

10.12289 

.44 

0 

fill    90796 

.15 

13874  •*"   0 

M. 

Cosine.  ;  PI"  ;  CotHiiir. 

1)1"   M. 

M    C(i<ino.   PI"   Cotan-r.   I>1"   1H  . 

54 


54° 


SINES  AND  TANGENTS. 


55' 


M. 

Sine. 

Dl" 

Tung. 

Dl" 

M. 

M. 

Sine. 

Dl" 

THUS.  . 

Dl" 

M. 

0 

1 

9.90796 
90805 

0.15 

1  S 

10.13874 
13900 

0.44 

A  A 

60 

59 

0 
1 

9.91336 
91345 

0.15 

1  S 

10.15477 
15504 

0.45 

60 
59 

2 

90814 

*  JLO 

13927 

•'±4 

58 

2 

91354 

•  1  0 

15531 

.40 

58 

3 
4 

90823 
90832 

.15 
.15 

13954 
13980 

.44 

.44 

57 
56 

3 
4 

91363 
91372 

.15 
.15 

15558 
15585 

.45 
.45 

57 
56 

5 

90842 

.15 

14007 

.44 

55 

5 

91381 

.15 

15612 

.45 

55 

6 

90851 

.15 

1  E 

14033 

.44 

A  A 

54 

6 

91389 

.15 

i  ^ 

15639 

.45 

A  f\ 

54 

7 

90860 

.10 

14060 

.44 

53 

7 

91398 

.  10 

15666 

.40 

53 

8 

90869 

.15 

-|  r 

14087 

.44 

A  A 

52 

8 

91407 

.15 

1  c 

15693 

.45 

A  fi 

52 

9 

90878 

.10 
1  Pi 

14113 

.44 

A  -1 

51 

9 

91416 

.10 

1  1 

15720 

.40 

A  f\ 

51 

10 

90887 

.10 

14140 

.44 

50 

10 

91425 

.10 

15746 

.40 

50 

11 

9.  90896 

.15 

1  c 

10.14166 

.44 

A  A 

49 

11 

9.91433 

.15 
i  R 

10.15773 

.45 

A  Pi 

49 

12 

90906 

.  10 

14193 

.44 

48 

12 

9!442 

.  I  0 

15800 

.40 

A  F; 

48 

13 

90915 

.15 

14220 

.44 

47 

13 

91451 

.15 

15827 

.40 

47 

14 

90924 

.15 

14246 

.44 

46 

14 

91460 

.15 

15854 

.45 

46 

15 

90933 

.15 

14273 

.44 

45 

15 

91469 

.15 

15881 

.45 

45 

16 

90942 

.15 

i  ^ 

14300 

.44 

A  A 

44 

16 

91477 

.15 

15908 

.45 

A  Pi 

44 

17 

90951 

.10 

14326 

.44 

43 

17 

91486 

.15 

15935 

.40 

43 

18 

90960 

.15 

14353 

.44 

42 

18 

91495 

.15 

15962 

.45 

42 

19 

90969 

.15 

1  Pi 

14380 

.44 

A  A 

41 

19 

91504 

.15 

-|  e 

15989 

.45 

A  *\ 

41 

20 

90978 

.10 

1  Pi 

14406 

.44 

A  t 

40 

20 

91512 

.10 

16016 

.40 

A  Pi 

40 

21 

9.90987 

.10 

10.14433 

.44 

39 

21 

9.91521 

.15 

10.16043 

.40 

39 

22 

90996 

.15 

1  r 

14460 

.44 

38 

22 

91530 

.15 

16070 

.45 

38 

23 
24 

91005 
91014 

.10 

.15 

14486 
14513 

.44 
.44 

37 
36 

23 
24 

91538 
91547 

.15 
.15 

16097 
16124 

.45 
.45 

37 
36 

25 

91023 

.15 

14540 

.44 

35 

25 

91556 

.15 

16151 

.45 

35 

26 

91033 

.15 

14566 

.44 

34 

26 

91565 

.15 

16178 

.45 

34 

27 

91042 

.15 

14593 

.45 

33 

27 

91573 

.14 

16205 

.45 

33 

28 

91051 

.15 

14620 

.45 

32 

28 

91582 

.14 

16232 

.45 

32 

29 

91060 

.15 

14646 

.45 

31 

29 

91591 

.14 

16260 

.45 

31 

30 

91069 

.15 

14673 

.45 

30 

30 

91599 

.14 

16287 

.45 

30 

31 

9.91078 

.15 

10.14700 

.45 

29 

31 

9.91608 

.14 

10.16314 

.45 

29 

32 

91087 

.15 
i  \ 

14727 

.45 

A  c 

28 

32 

91617 

.14 

16341 

.45 

A  Pi 

28 

33 
34 

91096 
91105 

.10 

.15 

14753 

14780 

.40 

.45 

27 
26 

33 
34 

91625 
91634 

.14 
.14 

16368 
16395 

.40 
.45 

27 
26 

35 

91114 

.15 
1  *\ 

14807 

.45 

A  Pi 

25 

35 

91643 

.14 

16422 

.45 

A  Pi 

25 

36 

91123 

.10 

i  £ 

14834 

.40 

24 

36 

91651 

.14 

16449 

.40 

24 

37 

91132 

.10 

1  £» 

14860 

.45 

A  Ci 

23 

37 

91660 

.14 

16476 

.45 

23 

38 

91141 

.10 

14887 

.40 

22 

38 

91669 

.14 

16503 

.45 

22 

39 

91149 

.15 

14914 

.45 

21 

39 

91677 

.14 

16530 

.45 

21 

40 

91158 

.15 

14941 

.45 

20 

40 

91686 

.14 

16558 

.45 

20 

41 

9.91167 

.15 

1  Pi 

10.11967 

.45 

t  r- 

19 

41 

9.91695 

.14 

10.16585 

.45 

19 

42 

91176 

.10 

14994 

.40 

18 

42 

91703 

.14 

16612 

.45 

18 

43 

91185 

.15 

i  fi 

15021 

.45 

17 

43 

91712 

.14 

16639 

.45 

17 

44 

91194 

.10 

15048 

.45 

16 

44 

91720 

.14 

16666 

.45 

16 

45 

91203 

.15 

i  Pi 

15075 

.45 

15 

45 

91729 

.14 

16693 

.45 

15 

46 

91212 

.1  0 

15101 

.45 

14 

46 

91738 

.14 

16720 

.45 

14 

47 

91221 

.15 
1  ^ 

15128 

.45 

13 

47 

91746 

.14 

16748 

.45 

13 

48 

91230 

.10 

•f  r 

15155 

.45 

12 

48 

91755 

.14 

16775 

.45 

12 

49 

91239 

.10 

15182 

.45 

11 

49 

91763 

.14 

16802 

.45 

11 

50 

91248 

.15 

1  c 

15209 

.45 

10 

50 

91772 

.14 

16829 

.45 

10 

51 

9.91257 

.10 

10.15236 

.45 

9 

51 

9.91781 

.14 

10.16856 

.45 

9 

52 

91266 

.15 

15262 

.45 

8 

52 

91789 

.14 

16883 

.45 

8 

53 

91274 

.15 

i  *i 

15289 

.45 

7 

53 

91798 

.14 

16911 

.45 

7 

54 

91283 

.  10 

15316 

.45 

6 

54 

91806 

.14 

16938!  '*? 

6 

55 

91292 

.15 

15343 

.45 

5 

55 

91815 

.14 

16965 

.40 

5 

56 

91301 

.15 

1  e 

15370 

.45 

4 

56 

91  823 

.14 

16992 

.45 

4 

57 
58 

91310 
91319 

.10 

.15 

i  Pi 

15397 
15424 

.45 
.45 

3 

2 

57 
58 

91832 
91840 

.14 
.14 

17020 
17047 

•45 
.45 

3 
2 

59 

91328 

.10 

15450 

.45 

1 

59 

91849 

.14 

17074 

.45 

1 

60 

91336 

.15 

15477 

.45 

0 

60 

91857 

.14 

17101 

.45 

0 

M. 

Cosine. 

Dl"  |  Cotang.   Dl" 

M. 

M. 

Cosine. 

Dl" 

Cot  a  MS. 

Dl" 

M. 

55 


34° 


56' 


TABLE  IV.— LOGARITHMIC 


57° 


31. 

Si  no.  i  Dl" 

Tans,'.  ;  Dl" 

M. 

M. 

Sine. 

Dl" 

Tung. 

1)1" 

M. 

0 
1 

9.91857' 
91866 

0.14 

10.17101 
17129 

0.45 

60 
59 

0 
1 

9.92359 
92367 

0.14 

10.18748 
18776 

0.46 

60 

59 

2 

91874 

.14 

17156 

.45 

A  C 

58 

2 

92376 

.14 

1  A 

18804 

.46 

A  £ 

58 

3 

91883 

17183 

.40 

57 

3 

92384 

.  14 

18831 

.40 

57 

4 

91891 

.14 

17210 

.45 

56 

4 

92392 

.14 

18859 

.46 

56 

5 

91900 

.14 

17238 

.45 

55 

5 

92400 

.14 

18887 

.46 

55 

6 

91908 

.14 

17265 

.45 

54 

6 

92408 

.14 

18914 

.46 

54 

7 

91917 

.14 

17292 

.45 

53 

7 

92410 

.14 

18942 

.46 

53 

8 

91925 

.14 

17319 

.45 

52 

8 

92425 

.14 

18970 

.46 

52 

9 

91934 

.14 

17347 

.46 

51 

9 

92433 

.14 

18997 

.46 

51 

10 

91942 

.14 

i  * 

17374 

.46 

40 

50 

10 

92441 

.14 

19025 

.46 

A  /» 

50 

11 

9.91951 

.1-4 

10.17401 

.40 

49 

11 

9.92449 

10.19053 

.40 

49 

12 

91959 

.14 

1  4 

17429 

.46 

1  A 

48 

12 

92457 

.14 

19081 

.46 

A  a 

48 

13 
14 

91968 
91976 

.14 

.14 

17456 

17483 

.-40 

.46 

47 
46 

13 
14 

92465  •'* 
92473  '!* 

19108 
19136 

.40 
.46 

47 
46 

15 

91985 

.14 

17511 

.46 

45 

15 

92482 

.14 

19164 

.46 

45  - 

16 

91993 

.14 

17538 

.46 
\  A. 

44 

16 

92490 

.14 

19192 

.46 

A  A 

44 

17 

92002 

17565 

.40 

43 

17 

92498 

19219 

.40 

43 

18 

19 

92010 
92018 

.14 
.14 

17593 
17620 

.46 
.46 

A  £t 

42 

41 

18 
19 

92506 
92514 

.14 
.14 

19247 
19275 

.46 
.46 

42 
41 

20 

92027 

.14 

17648 

.40 

40 

20 

92522 

.13 

19303 

.46 

40 

21 

9.92035 

.14 

10.17675 

.46 
.1  & 

39 

21 

9.92530 

.13 

-I  0 

10.19331 

.46 

A  £ 

39 

22 

92044 

17702 

.40 

38 

22 

92538 

.10 

19358 

.40 

38 

23 

92052 

.14 

17730 

.46 

A  A 

37 

23 

92546 

.13 

1  o 

19386 

.46 

37 

24 

92060 

17757 

.40 

\  fi 

36 

24 

92555 

.  1  6 

1  o 

19414 

.46 

A  C 

36 

25 

92069 

17785 

.40 
dfi 

35 

25 

92563 

•lo 

1  Q 

19442 

.40 

A  A 

35 

26 

92077 

17812 

.40 

34- 

26 

92571 

•  lo 

19470 

.40 

34 

27 

92086 

.14 

1  A 

1  7839 

.46 

40 

33 

27 

92579 

.13 

1  o 

19498 

.46 

A  £ 

33 

28 
29 

92094 
92102 

•  14 

.14 

17867 
17894 

.40 

.46 

32 
31 

28 
29 

92587 
92595 

.  1  O 

.13 

19526 
19553 

.4o 
.46 

32 
31 

30 

92111 

.14 

17922 

.46 

30 

30 

92603 

.13 

19581 

.46 

30 

31 

9.92119 

.14 

10.17949 

.46 
j.fi 

29 

31 

9.92611 

.13 

1  Q 

10.19609 

.46 

A  A 

29 

32 

92127 

17977 

.40 

28 

32 

92619 

•  1  •> 

19637 

.4o 

28 

33 

92136 

.14 

18004 

.46 

1  A 

27 

33 

92627 

.13 
-i  «> 

19665 

.47 

27 

34 

92144 

18032 

.40 
4fi 

26 

34 

92635 

.1  o 

1  9 

19693 

.47 

A  7 

26 

35 

92152 

18059 

.40 

25 

35 

92643 

•  13 

19721 

.4( 

25 

36 

92161 

.14 

18087 

.46 

i  A 

24 

36 

92651 

.13 

1  O 

19749 

.47 

24 

37 

92169 

18114 

.40 

23 

37 

92659 

•  lo 

19777 

.47 

23 

38 

92177 

.14 

18142 

.46 

22 

38 

92667!  'JJ 

19805 

.47 

22 

39 

92186 

.14 

18169 

.46 

1  A 

21 

39 

92675!  >J* 

19832 

.47 

A  7 

21 

40 

92194 

18197 

.40 

20 

40 

92683]  '{J 

19860 

.47 

20 

41 

9.92202 

.14 

10.18224 

.46 

A  A 

19 

41 

9.92691  i  *fj 

10.19888 

.47 

19 

42 

92211 

18252 

.40 

18 

42 

926991  *'„ 

199  Hi 

.47 

18 

43 

92219 

.14 

18279 

.46 

17 

43 

92707 

.Id 

19944 

.47 

17 

44 

92227 

.14 

18307 

.46 

16 

44 

92715 

.13 

19972 

.47 

16 

45 

46 

92235 
92244 

.14 
.14 

18334 
18362 

.46 
.46 

i  £ 

15 
14 

45 

46 

92723 
92731 

.13 
.13 

1  *.} 

20000 
20028 

.47 

.47 

15 
14 

47 

92252 

.14 

18389 

.40 

1  f* 

13 

47 

92739 

.  l  •> 
i  •} 

20056 

.47 

A  7 

13 

48 

92260 

18417 

.40 

iO 

12 

48 

92747 

.lo 
i  S 

20084 

A  I 

A  *7 

12 

49 

92269 

18444 

.40 

11 

49 

92755 

.  l  •> 

20112 

.4i 

11 

50 

92277  j  ' 

18472 

.46 

4« 

10 

50 

92763 

.13 
1  3 

20140 

.47 

47 

10 

51 
52 

9.92285 
92293 

.  i-± 
.14 

10.18500 

18527 

•  TcO 

.46 

9 
8 

51 

52 

9.92771 
92779 

.10 
.13 

10.20168 
20196 

•4  1 

.47 

9 

8 

53 

92302 

.14 

18555 

.46 

to 

7 

53 

92787 

.13 

1  *? 

20224 

.47 

A  7 

7 

54 
55 

92310 
92318 

.14 

18582 
18610 

.40 

.46 

4  A 

6 
5 

54 
55 

92795 
92803 

*  I  O 

.13 

1  9 

20253 
20281 

«4< 

.47 

I  *? 

6 
5 

56 

92326 

.14 

1  8638 

.40 

4 

56 

92810 

•  lo 

20309 

.4< 

4 

57 

92335 

•14 

18665 

.4fi 

3 

57 

92818 

.13 

20337 

.47 

3 

58 

9234'! 

.14 

18693 

.4(i 

2 

58 

92826 

.13 

20365 

.47 

2 

59 

92351 

.14 

18721 

.4fi 

1 

59 

92834  i  '** 

2039.°, 

.47 

1 

80 

92359 

.14 

18M8 

.46 

0 

80 

92842  *™ 

20421 

.47 

0 

M. 

Cosine. 

nr 

Cot;tll2. 

T)l" 

M. 

M.  cosinf.  ni" 

('otanz. 

Dl" 

M. 

32° 


SINES  AND    TANGENTS. 


59* 


31. 

Sine. 

1)1" 

Tang. 

Dl" 

M. 

M. 

Sine. 

Dl" 

Tang. 

Dl" 

M. 

0 
1 

9.92842 
02850 

0.13 

10.20421 
20449 

0.47 

60 
59 

0 

1 

9.93307 
93314 

0.13 

10.22123 
22151 

0.48 

60 
5'J 

2 

92858 

.13 

20477 

.47 

58 

2 

93322 

.13 

22180 

.48 

58 

3 

92866 

.13 

1  o 

20505 

.47 

57 

3 

93329 

.13 

1  9 

22209 

.48 

A  Q 

57 

4 

92874 

.10 

20534 

.47 

56 

4 

93337 

.  lo 

22237 

.48 

56 

5 
6 

92881 
92889 

.13 
.13 

20562 
20590 

.47 

.47 

55 
54 

5 

6 

93344 
93352 

.13 
.13 

22266 
22294 

.48 
.48 

55 
54 

92897 

.13 

20618 

.47 

53 

7 

93360 

.13 

22323 

.48 

53 

8 

92905 

.13 

20646 

.47 

52 

8 

93367 

.13 

22352 

.48 

52 

9 

92913 

.13 

20674 

.47 

51 

9 

93375 

.13 

22381 

.48 

51 

10 

92921 

.13 

20703 

.47 

50 

10 

93382 

.13 

22409 

.48 

50 

11 

9.92929 

.13 

10.20731 

.47 

49 

11 

9.93390 

.13 

10.22438 

.48 

49 

12 

92936  '!„ 

20759 

.47 

48 

12 

93397 

.13 

22467 

.48 

48 

13 

92944 

.lo 

20787 

.47 

47 

13 

93405 

.13 

22495 

.48 

47  • 

14 

92952 

.13 

20815 

.47 

46 

14 

93412 

.13 

22524 

.48 

46 

15 

92960 

.13 

20844 

.47 

45 

15 

93420 

.13 

22553 

.48 

45 

16 

92968 

.13 

20872 

.47 

44 

16 

93427 

.13 

22582 

.48 

44 

17 

92976 

.13 

20900 

.47 

43 

17 

93435 

.13 

22610 

.48 

43 

18 

92983 

.13 

20928 

.47 

42 

18 

93442 

.13 

22639 

.48 

42 

19 

92991 

.13 

20957 

.47 

41 

19 

93450 

.12 

22668 

.48 

41 

20 

92999 

.13 

20985 

.47 

40 

20 

93457 

.12 

22697 

.48 

40 

21 

9.93007 

.13 

10.21013 

.47 

39 

21 

9.93465 

.12 

10.22726 

.48 

39 

22 

93014 

.13 

21041 

.47 

38 

22 

93472 

.12 

22754 

.48 

38 

23 

93022 

.13 

21070 

.47 

37 

23 

93480 

.12 

22783 

.48 

37 

24 

93030 

.13 

21098 

.47 

36 

24 

93487 

.12 

22812 

.48 

i  Q 

36 

25 

93038 

.13 

21126 

.47 

35 

25 

93495 

.12 

22841 

.48 

35 

26 

93046 

.13 

21155 

.47 

34 

26 

93502 

.12 

22870 

.48 

A  Q 

34 

27 

93053 

.13 

21183 

.47 

33 

27 

93510 

.12 

22899 

.48 

33 

28 

93061 

.13 

21211 

.47 

32 

28 

93517 

.12 

22927 

.48 

32 

29 

93069 

.13 

21240 

.47 

31 

29 

93525 

.12 

22956 

.48 

A  Q 

31 

30 

93077 

.13 

21268 

.47 

30 

30 

93532 

.12 

22985 

.48 

30 

31 

32 

9.93084 
93092 

.13 
.13 

10.21296 
21325 

.47 
.47 

29 
28 

31 

32 

9.93539 
93547 

.12 
.12 

10.23014 
23043 

.48 
.48 

29 

28 

33 

93100 

.13 

1  o 

21353 

.47 

A  *7 

27 

33 

93554 

.12 

23072 

.48 

A  Q 

27 

34 

93108 

•la 

21382 

.47 

26 

34 

93562 

.12 

23101 

«4o 

26 

35 

93115 

.13 

21410 

.47 

25 

35 

93569 

.12 

23130 

.48 

25 

36 

93123 

.13 

21438 

.47 

24 

36 

93577 

.12 

23159 

.48 

24 

37 

93131 

.13 

21467 

.47 

23 

37 

93584 

.12 

23188 

.48 

23 

38 

93138 

.13 

21495 

.47 

22 

38 

93591 

.12 

23217 

.48 

22 

39 

93146 

.13 

21524 

.47 

21 

39 

93599 

.12 

23246 

.48 

21 

40 

93154 

.13 

1  O 

21552 

.47 

20 

40 

93606 

.12 

23275 

.48 

20 

41 

9.93161 

.lo 

10.21581 

.47 

19 

41 

9.93614 

.12 

10.23303 

.48 

19 

42 

93169 

.13 

1  Q 

21609 

.47 

4»7 

18 

42 

93621 

•12 

1  n 

23332 

.48 

40 

18 

43 

93177 

•  J  0 

21637 

7 

17 

43 

93628 

•  LZ 

23361 

.48 

17 

44 

93184 

.13 

1  Q 

21666 

.47 

4*- 

16 

44 

93636 

.12 

1  Q 

23391 

.48 

.JO 

16 

45 

93192 

.lo 

21694 

j 

15 

45 

93643 

•  Iz 

23420 

.48 

15 

46 

93200 

.13 

1  Q 

21723 

.47 

/I  T 

14 

46 

93650 

.12 
i  o 

23449 

.48 

A  Q 

14 

47 

93207 

.lo 

21751 

.47 

13 

47 

93658 

.12 

23478 

.48 

13 

48 

93215 

.13 

1  O 

21780 

.48 

A  Q 

12 

48 

93665 

.12 

1  rt 

23507 

.48 

1  Q 

12 

49 
50 

93223 
93230 

•Jo 
.13 

21808 
21837 

•4o 
.48 

11 
10 

49 
50 

93673 
93680 

•  I  2 
.12 

23536 
23565 

.4o 

.48 

11 
10 

51 

9.93238 

.13 

10.21865 

.48 

9 

51 

9.93687 

.12 

10.23594 

.48 

9 

52 

93246 

.13 

21894 

.48 

8 

52 

93695 

.12 

23623 

.48 

8 

53 

93253 

.13 

10 

21923 

.48 
40 

7 

53 

93702 

.12 
1  9 

23652 

.48 
4Q 

7 

54 

93261 

.1  o 

21951 

•  rto 

6 

54 

93709 

.  1  Zt 

23681 

.4y 

6 

55 

93269 

.13 

1  o 

21980 

.48 

5 

55 

93717 

.12 

23710 

.49 

5 

56 

93276 

.13 

22008 

.48 

4 

56 

93724 

.12 

23739 

.49 

4 

57 

93284 

.13 
1  3 

22037 

.48 

40 

3 

57 

93731 

.12 

1  9 

23769 

.49 

AQ 

3 

58 

93291 

•  lo 
1  q 

22065 

:8 

AQ 

2 

58 

93738 

•  1  4 
1  o 

23798 

.4  y 

A  O 

2 

59 

93299 

.lo 
1  O 

22094 

.48 

A  Q 

1 

59 

93746 

•  \  1 

1  O 

23827 

.4y 

1 

60 

93307 

.lo 

22123 

.4c 

0 

60 

93753 

.12 

23856 

.49 

0 

M. 

Cosine. 

Dl" 

Gotang. 

Dl" 

M. 

M. 

Cosine. 

Dl" 

Cot:uur. 

Dl" 

M. 

Slc 


57 


6O° 


TABLE  IV.— LOGARITHMIC 


61 


M. 

Sine,  i  Dl" 

Tang.   Dl" 

M. 

M. 

Sine. 

1)1" 

Tang. 

Dl" 

M. 

0 

9.93753L  L 

10.  23856  1 

60 

0 

9.94182 

10.25625 

60 

1 

93760 

U.iZ 

23885  °'*l 

59 

1 

94189 

0.12 

In 

25655 

0.50 

r  ,i 

59 

2 

93768 

.12 

23914  -Jr 

58 

2 

94196 

2 

25684 

.OU 

58 

3 

93775 

.12 

23944!  -JJ 

57 

3 

94203 

.12 

25714 

.50 

57 

4 

93782 

.12 

1  O 

23973  i  •*! 

56 

4 

94210 

.12 

1  o 

25744 

.50 

C  A. 

56 

5 

93789 

.12 

24002 

.IV 

55 

5 

94217 

.  1  2 

25774 

.OU 

55 

6 

7 

93797 
93804 

.12 
.12 

24031 
24061 

.49 
.49 

54 
53 

6 

7 

94224 

94231 

.12 
.12 

25804 
25834 

.50 
.50 

C  A 

54 
53 

8 

938  U 

.12 

24090 

.49 

52 

8 

94238 

.12 

25863 

.50 

52 

9 

93819 

.12 

1  O 

24119 

.49 

51 

9 

94245 

.12 

25893 

.50 

en 

51 

10 
11 

93826 
9.93833 

.  IZ 
.12 

1  A 

24148 
10.24178 

!49 

50 
49 

10 
11 

94252 
9.94259 

!l2 
i  a 

25923 
10.25953 

.oU 
.50 

50 
49 

12 

93840 

.12 

24207 

.49 

48 

12 

94266  •" 

25983 

.50 

48 

13 

93847 

.12 

24236 

.49 

47 

13 

94273  '" 

26013 

.50 

47 

14 

93855 

.12 

24265 

.49 

46 

14 

94279  '** 

26043 

.50 

46 

15 
16 
17 

93862 
93869 
93876 

.12 
.12 
.12 

24295 
24324 
24353 

.49 

.49 
.49 

45 
44 
43 

15 
16 
17 

94286  '" 
94293  '  ; 
94300  ';* 

26073 
26103 
26133 

.50 
.50 
.50 

45 
44 
43 

18 

93884 

.12 

24383 

.49 

42 

18 

94307!  •" 

26163 

.50 

42 

19 

93891 

.12 

1  O 

24412 

.49 

1  O 

41 

19 

94314  l  Tf 

26193 

.50 

41 

20 

93898 

.12 

24442 

.4» 

40 

20  i  943211  *ff 

26223 

.50 

40 

21 

9.93905 

.12 

10.24471 

.49 

39 

21  9.94328  '{, 

10.26253 

.50 

39 

22 

93912 

.12 

24500 

.49 

38 

22  i  94335,  ,! 

26283 

.50. 

38 

23 

93920 

.12 

24530 

.49 

37 

23    94342  '} 

26313 

.50 

37 

24 

93927 

.12 

24559 

A  A 

36 

24  j  94349  ''! 

26343 

.50 

36 

25 

93934 

.12 

24589 

.49 

35 

25    94355 

.11 

26373 

.50 

35 

26 

93941 

.12 

24618 

.49 

34 

26 

94362 

.11 

26403 

.50 

r  A 

34 

27 

93948 

.12 

24647 

4  A 

33 

27 

94369  -Jl 

26433 

•  OU 

33 

28 

93955  1  •" 

24677 

.49 

32 

28 

94376  '}; 

26463 

.50 

32 

29 

93963  'Jo 

24706 

.49 

31 

29 

94383)  '}! 

26493 

.50 

tn 

31 

30 

93970 

1  9 

24736 

A  O 

30 

30 

94390  '" 

26524 

•  Ol/ 
c  A 

30 

31 

9.93977 

•  12 

10.24765 

••*» 

29 

31 

9.943971  ' 

10.26554 

.OU 

29 

32 

93984 

.12 

1  rt 

24795 

.49 

28 

32 

94404   „ 

26584 

.50 

C  A 

28 

33 
34 

93991 
93998 

.!/ 

.12 
1  9 

24824 
24854 

'.49 

27 
26 

33 
34 

94410 
94417 

.11 

26614 
26644 

.OU 

.5-0 

CA 

27 
26 

35 

94005 

•  12 

24883 

Ay 

25 

35 

94424 

n 

•  1  1 

26674 

.OU 

25 

36 

94012 

1  O 

24913 

A  A 

24 

36 

94431 

.11 

26705 

.50 

24 

37 

94020 

.12 

24942 

.49 

23 

37 

94438 

.11 

26735 

.50 

23 

38 
39 

94027 
94034 

.12 
.12 

24972 
25002 

^49 

22 
21 

38 
39 

94445 
94451 

.11 

.11 

26765 
26795 

.50 
.50 

22 
21 

40 

94041 

.12 

25031 

A  A 

20 

40 

94458 

.11 

26825 

.50 

20 

41 

9.94048 

.12 

19 

10.25061 

.49 
4Q 

19 

41 

9.94465 

.11 

10.26856 

.50 

EA 

19 

42 

94055 

.12 

1  A 

25090 

.4» 

18 

42 

94472!  "'I 

2688« 

•  OU 

C  A 

18 

43 

94062|  '!A 

25120 

in 

17 

43 

94479 

.1  1 

26916 

.OU 

17 

44 

94069 

.1^ 

25149 

.49 

16 

44 

94485 

.11 

26946 

.50 

16 

45 

94076 

.12 

-I  A 

25179 

.49 

\(\ 

15 

45 

94492 

.11 

26977 

.50 

e  T 

15 

46 

94083 

.12 

1  A 

25209 

.4W 

14 

46 

94499 

27007 

•01 

14 

47 

94090 

.12 

1  A 

25238 

.49 

13 

47 

94506 

.11 

27037 

.51 

c-i 

13 

48 

94098 

.12 

25268 

A  (i 

12 

48 

94513 

-t  -i 

27068 

.01 

c  -i 

12 

49 

94105 

1  O 

25298 

.4» 

A  O 

11 

49 

94519 

.11 

27098 

.01 

r  1 

11 

50 

94112 

.  12 

25327 

.4y 

10 

50 

94526 

.11 

27128 

.01 

r  i 

10 

51 

9.94119 

.12 

10.25357 

-49 

9 

51 

9.94533 

. 

10.27159 

.01 

9 

52 

94126 

.12 

25387 

.50 

8 

52 

94540 

.11 

27189 

.51 

8 

53 

94133 

.12 

1  9 

25417 

.50 

C  A 

7 

53 

94546 

.11 

27220 

.51 

7 

54 

94140 

.12 

25446 

.OU 

6 

54 

94553 

27250 

. 

6 

55 

94147 

.12 

25476 

.50 

5 

55 

94560 

.11 

27280 

.51 

5 

56 

94154 

.12 

25506 

.50 

4 

56 

94567 

.11 

27311 

.51 

4 

57 

94161 

.12 

25535 

.50 

3 

57 

94573 

.11 

27341 

.51 

3 

58 

94168 

.12 

1  9 

25565 

.50 

2 

58 

94580 

.11 

27372 

.51 

£L1 

2 

59 

94175 

.12 
1  n 

25595 

.OU 

1 

59 

94587 

27402 

.01 

1 

60 

94182 

.12 

25625 

.50 

0 

60 

94593 

.11 

27433 

.51 

0 

M. 

Cosine. 

Dl" 

Cotang. 

Dl" 

M. 

M. 

Cosine. 

Dl" 

Cotang. 

D\" 

M. 

29° 


SINES  AND    TANGENTS. 


63C 


M. 

Sine. 

D." 

Tang. 

Dl" 

M. 

M. 

Sine. 

Dl" 

Tang. 

DI" 

M. 

0 
1 

9.94593 
94600 

0.11 

10.27433 
27463 

0.51 

60 
59 

0 
1 

9.94988 
94995 

0.11 

10.29283 
29315 

0.52 

60 
59 

2 

94607 

.11 

27494J  '?! 

58 

2 

95001 

.11 

29346 

.52 

58 

3 

94614 

.11 

27524 

.01 

57 

3 

95007 

.11 

29377 

.52 

57 

4 

94620 

.11 

27555 

.51 

56 

4 

95014 

.1  1 

29408 

.52 

co 

56 

5 

94627 

.11 

27585 

.51 

55 

5 

95020 

.11 

29440 

.OZ 

55 

6 

94634 

.11 

27616 

.51 

54 

6 

95027 

.11 

29471 

.52 

54 

7 

94640 

.11 

27646 

.51 

53 

7 

95033 

.11 

29502 

.52 

53 

8 

94647 

.11 

27677 

.51 

52 

8 

95039 

.11 

29534 

.52 

52 

9 

94654 

.11 

27707 

.51 

51 

9 

95046 

.11 

29565 

.52 

51 

10 

94660 

.11 

27738 

.51 

50 

10 

95052 

.11 

29596 

.52 

50 

11 

9.94667 

.11 

10.27769 

.51 

49 

11 

9.95059 

.11 

10.29628 

.52 

co 

49 

12 

94674 

.11 

27799 

.51 

48 

12 

95065 

.11 

29659 

.OZ 

48 

13 

94680 

.11 

27830 

.51 

47 

13 

95071 

.11 

29691 

.52 

47 

14 

94687 

.11 

27860 

.51 

46 

14 

95078 

.11 

29722 

.52 

46 

15 

94694 

.11 

27891 

.51 

45 

15 

95084 

.11 

29753 

.52 

45 

16 

94700 

.11 

27922 

.51 

44 

16 

95090 

.11 

29785 

.52 

44 

17 

94707 

.11 

27952 

.51 

43 

17 

95097 

.11 

29816 

.52 

co 

43 

18 

94714 

.11 

27983 

.51 

42 

18 

95103 

.11 

29848 

.OZ 

42 

19 

94720 

.11 

28014 

.51 

41 

19 

95110 

.11 

29879 

.52 

41 

20 

94727 

.11 

28045 

.51 

40 

20 

95116 

.11 

29911 

.52 

40 

21 

9.94734 

.11 

10.28075  -JJI 

39 

21 

9.95122 

.11 

10.29942 

.53 

39 

22 

94740 

.11 

•  28106  '£} 

38 

22 

95129 

.11 

29974 

.53 

r  o 

38 

23 

94747 

.11 

28137  '{J 

37 

23 

95135 

.11 

30005 

.5o 

CO 

37 

24 

94753 

.11 

28167  'r,1 

36 

24 

95141 

.11 

30037 

.Oo 

36 

25 

94760 

.11 

28198 

.01 

35 

25 

95148 

.11 

30068 

.53 

35 

26 

94767 

.11 

28229 

.51 

34 

26 

95154 

.11 

30100 

.53 

34 

27 

94773 

.11 

28260  '?! 

33 

27 

95160 

.11 

30132 

.53 

CO 

33 

28 

94780 

.11 

28291 

.01 

32 

28 

95167 

.11 

30163 

.00 

32 

29 

94786 

.11 

28321 

.51 

31 

29 

95173 

.11 

30195 

.53 

31 

30 

94793  -!! 

28352 

.51 

30 

30 

95179 

.11 

30226 

.53 

30 

31 

9.94799 

.ii 

10.28383 

.51 

29 

31 

9.95185 

.10 

10.30258 

.53 

CO 

29 

32 

94806 

.11 

28414 

.51 

28 

32 

95192 

'•}J   30290 

.00 

28 

33 

94813 

.11 

28445 

.51 

27 

33 

95198 

.1U 

30321 

.53 

27 

34 

94819 

.11 

28476 

.51 

26 

34 

95204 

.10 

30353 

.53 

26 

35 

94826 

.11 

28507 

.51 

25 

35 

95211 

.10 

30385 

.53 

25 

36 

94832 

.11 

28538 

.52 

24 

36 

95217 

.10 

30416 

.53 

24 

37 

94839 

.11 

28569 

.52 

23 

37 

95223 

.10 

30448 

.53 

23 

38 

94845 

.11 

28599 

.52 

CO 

22 

38 

95229 

.10 

-I  A 

30480 

.53 

c  o 

22 

39 

94852 

28630 

.oz 

21 

39 

95236 

.11) 

30512 

.0.1 

21 

40 

94858 

.11 

28661 

.52 

20 

40 

95242 

.10 

30543 

.53 

20 

41 

9.94865 

.11 
i  -i 

10.28692 

.52 

19 

41 

9.95248 

.10 

1  A 

10.30575 

.53 

C  0 

19 

42 

94871 

.11 

28723 

.52 

18 

42 

95254 

•  1U 

30607 

.Oo 

18 

43 

94878 

.11 

28754  j  '?£ 

17 

43 

95261 

.10 

1  A 

30639 

.53 

r  o 

17 

44 

94885 

28785!  *?* 

16 

44 

95267 

•  1U 

30671 

.Do 

16 

45 

94891 

.11 

28816 

.02 

15 

45 

95273 

.10 

30702 

.53 

15 

46 

94898 

.11 

28847 

.52 

14 

46 

95279 

.10 

30734 

.53 

14 

47 

94904 

.11 

28879 

.52 

13 

47 

95286 

.10 
in 

30766 

.53 

CO 

13 

48 

94911 

28910 

.52 

12 

48 

95292 

•  lu 

30798 

.Oo 

12 

49 

94917 

.11 

28941 

.52 

11 

49 

95298 

.10 

30830 

.53 

11 

50 

94923 

.11 

28972 

.52 

10 

50    95304 

.10 

30862 

.53 

10 

51 

9.94930 

.11 

10.29003 

.52 

c  n 

9 

51  19.95310 

.10 

1  A 

10.30894 

.53 

CO 

9 

52 

94936 

29034 

•02 

8 

52 

95317 

.1U 

30926 

.Do 

8 

53 

94943 

.11 

29065 

.52 

co 

7 

53 

95323 

.10 

30958 

.53 

CO 

7 

54 

94949 

29096 

.OZ 

6 

54 

95329 

.10 

30990 

.Do 

6 

55 

94956 

.11 

29127 

.52 

5 

55 

95335 

.10 

31022 

.53 

5 

56 

94962 

.11 

29159 

.52 

co 

4 

56 

95341 

.10 

1  t\ 

31054 

.53 

CO 

4 

57 

94969 

29190 

.OZ 

3 

57 

95348 

.10 

31086 

.03 

3 

58 

94975 

.11 

29221 

.52 

2 

58 

95354 

.10 

31118 

.53 

2 

59 
60 

94982 
949S8 

.11 
.11 

29252)  •** 
29283  ' 

1 
0 

59 
60 

95360 
95366 

.10 
.10 

31150 
31182 

.53 
.53 

1 

0 

M.  j  Cosine. 

Dl" 

Ootang.   Dl" 

M. 

M. 

Cosine. 

Dl" 

Cotang. 

Dl" 

M. 

59 


20° 


TABLE  IV.— LOGARITHMIC 


M.   Sine. 

Dl"   Tang.   Di" 

M. 

M. 

Sine.   Dl"   Tung,  j  Dl" 

M. 

0  9.95366  A  ,n  10.31182  '„  .., 

60 

0  |9.95728  „  .„  10.33133 

60 

1 

95372 

u<{"    31214 

U.Od 

59 

1 

95733 

33166 

0.55  ,g 

2 

95378 

.10 

31246 

.53 

58 

2 

95739  '™ 

33199 

•??  58 

3 

95384 

.10 

31278 

.53 

57 

3 

95745 

.10 

33232 

'55  57 

4 

95391 

.  .10 

31310 

.54 

56 

4 

95751 

.10 

33265 

56 

5 

95397 

.10 

31342 

.54 

55 

5 

95737 

.10 

33298 

.55 

55 

6 

95403 

.10 

31.]  74 

.54 

54 

6 

95763 

.10 

33331 

.55 

54 

7 

95409 

.10 

31407 

.54 

53 

7 

95769 

.10 

33364 

.55 

53 

8 

95415 

.10 

31439 

.54 

52 

8 

95775 

.10 

33397 

.55 

52 

9 

95421 

.10 

31471 

.54 

51 

9 

95780 

.10 

33430 

.55 

51 

10 

95427 

.10 

31503 

.54 

50 

10 

95786 

.10 

33463 

.55 

50 

11 
12 

9.95434 
95440 

.10 
.10 

10.31535 
31568 

.54 
.54 

49 

48 

11 
12 

9.95792 
95798 

.10 
.10 

10.33497  •?? 
335301  •*• 

49 

48 

13 

95446 

.10 

31600 

.54 

47 

13 

95804 

.10 

33563  •?'? 

47 

14 
15 

95452 

95458 

.10 
.10 

31632 
31664 

.54 
.54 

46 
45 

14 
15 

95810 
95815 

.10 
.10 

33596 
33629 

.00 

.55 

46 
45 

16 

95464 

.10 

31697 

.54 

44 

16 

95821 

.10 

33663 

.55 

44 

17 

95470 

.10 

31729 

.54 

43 

17 

95827 

.10 

33696  •?? 

43 

18. 

95476 

.10 

1  A 

31761 

.54 

-A 

42 

18 

95833 

.10 
i  n 

33729  '*? 

42 

19 

95482 

.  IU 
1  0 

31794 

.04 

41 

19 

95839 

.10 
1  A 

337621  '2 

41 

20 

95488 

•  1  v 

31826 

40 

20 

95844 

.  J  U 

33796 

.uu 

40 

21 

9.95494 

.10 

10.31858 

.54 

39 

21 

9.95850 

.10 

10.33829 

.56 

39 

22 

95500 

.10 

31891 

.54 

38 

22 

95856 

.10 

33862 

.56 

38 

23 

95507 

.10 

31923 

.54 

37 

23 

95862 

.10 

33896!  '?? 

37 

24 

95513 

.10 

31956 

.54 

36 

24 

958(58  -J" 

33929 

.06 

36 

25 

95519 

.10 

31988 

.54 

35 

25 

95S73  \'!. 

33962 

.56 

35 

26 

95525 

.10 

32020 

.54 

34 

26 

95879 

.IU 

33996 

.56 

34 

27 

95531 

.10 

32053 

.54 

33 

27 

95885 

.10 

34029 

.56 

33 

28 

95537 

.10 

32085 

.54 

32 

28 

95891 

.10 

34063 

.56 

32 

29 

95543 

.10 

32118 

.54 

31 

29 

95897 

.10 

34096 

.56 

31 

30 

95549 

.10 

32150 

.54 

30 

30 

95902 

.10 

34130 

.56 

30 

31 

9.95555 

.10 

1  A 

10.32  1«3 

.54 

29 

31 

9.95908 

.10 

i  n 

10.34163 

.56 

p  n 

29 

32 

95561 

.10 

1  A 

32215 

.54 

28 

32 

95914 

.  1  0 
1  n 

34197 

.00 

c  rt 

28 

33 

95567 

.  1  0 

32248 

27 

33 

95920 

.10 

34230 

.00 

27 

34 

95573 

.10 

32281 

.54 

26 

34 

95925 

.10 

34264 

.56 

26 

35 

95579 

.10 

1  A 

32313 

.54 

25 

35 

95931 

.10 

1  0 

34297  '!?!? 

25 

36 

95585 

.1  U 

32346 

24 

36 

95937 

.  1  U 

34331  'JJ 

24 

;>7 

95591 

.10 

1  f\ 

32378 

.54 

23 

37 

95942 

.10 

1  A 

34364  -J5 

23 

38 

95597 

.10 

32411 

.54 

22 

38 

95948 

.  10 

34398 

.00 

22 

39 

95603 

.10 

32444 

.54 

21 

39 

95954 

.10 

34432 

.56 

21 

40 

95609 

.10 

32476 

.54 

20 

40 

95960 

.10 

34465 

.56 

20 

41 

9.95615 

.10 

10.32509 

.54 

19 

41 

9.95965 

.1  0 

10.34499 

.56 

19 

42 

95621 

.10 

32542 

.54 

18 

42 

95971 

.10 

34533 

.56 

18 

43 

95627 

.10 

1  A 

32574 

.55 

5e 

17 

43 

95977 

1  It  1 

34566 

.56 

c  /» 

17 

44 

95633 

.  10 

32607 

0 

16 

44 

95982 

.oy 

34600 

.00 

16 

45 

95639 

.10 

32640 

.55 

15 

45 

95988 

.09 

34634 

.56 

15 

46 

95645 

.10 

32673 

.55 

14 

46 

95994 

.09 

34667 

.56 

14 

47 

95651 

.10 

32705 

.55 

13 

47 

96000 

.09 

Afk 

34701 

.56 

13 

48 

95657 

.10 

32738 

.55 

12 

48 

96005 

.09 

34735 

.56 

12 

49 

95663 

.10 

32771 

.55 

11 

49 

96011 

.09 

34769 

.56 

11 

50 
51 

95668 
9.95674 

.10 
.10 

32804 
10.32837 

.55 
.55 

10 
9 

50 
51 

96017 
9.96022 

.09 
.09 

34803 
10.34836 

.56 
.56 

10 

9 

52 

95680 

.10 

32S69 

.55 

8 

52 

96028 

.09 

34870 

.56 

8 

53 

95686 

.10 

1  fi 

32902 

.55 

cc 

7 

53 

96034 

.09 
.09 

34904 

.56 
.56 

7 

54 

95692 

•  J  U 

32935 

.00 

6 

54 

96039 

34938 

6 

55 

95698 

.10 

32968 

.55 

5 

55 

96045 

.09 

A  A 

34972 

.57 

KT 

5 

56 

95704 

.10 

33001 

.55 

4 

56 

96050 

.09 

35006 

.07 

4 

57 

95710 

.10 

33034 

.55 

3 

57 

96056 

.09 

35040 

.57 

3 

58 

95716 

.10 

33067 

.55 

2 

58 

96062 

.09 

35074 

.57 

2 

59 

95722 

.10 

33100 

.55 

1 

59 

96067 

.09 

35108 

•o< 

1 

60 

95728 

.10 

1  0.33133 

.55 

0 

60 

96073 

.09 

35142 

.57 

0 

M. 

Cosine. 

Dl" 

Cotnnt. 

Dl" 

M. 

M.   Cosiiic.   Dl"   Cotang.   Di" 

M. 

25= 


66C 


SINES  AND  TANGENTS. 


67C 


M. 

Sine. 

Dl" 

Tang. 

Dl" 

M. 

M. 

Si  no. 

Dl" 

Tang. 

Dl" 

M. 

0 
1 

9.96073 
96079 

0.09 

10.35142 
35176 

0.57 

c  7 

60 

59 

0 
1 

9.96403 
96408 

0.09 

AQ 

J  0.372151 
37250 

0.59 

CQ 

60 
59 

2 
3 

96084 
96090 

.09 
.09 

An 

35210 
35244 

.Of 

.57 

57 

58 
57 

2 
3 

96413 
96419 

.uy 
.09 

no 

37285 
37320 

.oy 
.59 

K.O 

58 
57 

4 

96095 

.09 

Aft 

35278 

t 
c  7 

56 

4 

96424 

.uy 

OQ 

37355 

.oy 

CQ 

56 

5 

96101 

.uy 

35312 

.0  1 

55 

5 

96429 

.uy 

37391 

.oy 

55 

6 

7 
8 

96107 
96112 
96118 

.09 
.09 
.09 

35346 
35380 
35414 

.57 
.57 
.57 

K7 

54 
53 
52 

6 

7 
8 

96435 
96440 
96445 

.09 
.09 
.09 

OQ 

37426 
37461 
37496 

.59 
.59 
.59 
.59 

54 
53 
52 

9 

96123 

.uy 

35448 

•  0  1 

51 

9 

96451 

.uy 

37532 

51 

10 

96129 

.09 

Aft 

35483 

.57 

r  7 

50 

10 

96456 

.09 

Aft 

37567 

.59 

C  Q 

50 

11 

9.96135 

.09 

Aft 

10.35517 

.07 

c  n 

49 

11 

9.96461 

.uy 

OQ 

10.37602 

.oy 

CQ 

49 

12 

96140 

.uy 

35551 

.07 

48 

12 

96467 

.uy 

37638 

.oy 

48 

13 

96146 

.09 

OQ 

35585 

.57 

47 

13 

96472 

.09 

OQ 

37673 

.59 
fjQ 

47 

14 

96151 

.uy 

35619 

.57 

46 

14 

96477 

.uy 

Aft 

37708 

•  t>y 

eft 

46 

15 
16 

96157 
96162 

.09 
.09 

35654 

35688 

.57 
.57 

•S7 

45 
44 

15 
16 

96483 
96488 

.uy 
.09 
.09 

37744 
37779 

.oy 
.59 
.59 

45 

44 

17 

96168 

no 

35722 

.0  < 

t\7 

43 

17 

96493 

OQ 

37815 

P.O 

43 

18 

96174 

»uy 

AQ 

35757 

•Of 

42 

18 

96498 

.uy 

AQ 

37850 

.oy 

.59 

•42 

19 
20 

96179 
96185 

.uy 
.09 

OQ 

35791 
35825 

.57 

C7 

41 

40 

19 
20 

96504 
96509 

.uy 
.09 

AQ 

37886 
37921 

159 

F.Q 

41 

40 

21 

9.96190 

.uy 

OQ 

10.35860 

•  Of 

39 

21 

9.96514 

.uy 

.09 

10.37957 

.oy 
.59 

39 

22 

96196 

.uy 

AQ 

35894 

M 

38 

22 

96520 

OQ 

37992 

cq 

38 

23 

96201 

.uy 

Aft 

35928 

r  7 

37 

23 

96525 

•  Uy 
OQ 

38028 

.oy 

CQ 

37 

24 

96207 

.uy 

OQ 

35963 

.0  • 

R7 

36 

24 

96530 

.uy 
.09 

38064 

.oy 
.59 

36 

25 

96212 

.uy 

Aft 

35997 

•  O  i 

C  7 

35 

25 

96535 

Aft 

38099 

C  Q 

35 

26 

96218 

.uy 

OQ 

36032 

.07 

C7 

34 

26 

96541 

.uy 
.09 

38135 

.oy 
.59 

34 

27 

96223 

.uy 
OQ 

36066 

•  0  i 

CO 

33 

27 

96546 

38170 

33 

28 

96229 

.uy 

Aft 

36101 

.Oo 

c.Q 

32 

28 

96551 

Aft 

38206 

An 

32 

29 

96234 

.uy 

Aft 

36135 

.OO 

31 

29 

96556 

.uy 

Aft 

38242 

.OU 

31 

30 

96240 

.Uy 

36170 

.58 

30 

30 

96562 

.uy 

38278 

.60 

30 

31 

9.96245 

.09 
OQ 

10.36204 

.58 

K.O 

29 

31 

9.96567 

.09 
OQ 

10.38313 

.60 
(ill 

29 

32 

96251 

.uy 
OQ 

36239  •"" 

28 

32 

96572 

.uy 

AQ 

38349 

.ou 
fiO 

28 

33 

96256 

.uy 
no 

36274  |  *rj[ 

27 

33 

96577 

.uy 

Aft 

38385 

.OU 
An 

27 

34 

96262 

.uy 
.09 

36308  *r2 

26 

34 

96582 

.uy 

AQ 

38421 

.oU 
.60 

26 

35 

96267 

36343;  'r° 

25 

35 

96588 

.uy 

38456 

25 

36 
37 
38 
39 

96273 
96278 
96284 
96289 

.09 
.09 
.09 
.09 
09 

56377 
36412 
36447 
36481 

.08 

.58 
.58 
.56 

24 
23 
22 
21 

36 
37 
38 
39 

96593 
96598 
96603 
96608 

.09 
.09 
.09 
.09 

nn 

38492 
38528 
38564 
38600 

!eo 

.60 
.60 
fiO 

24 
23 
22 
21 

40 

96294 

36516  •"" 

20 

40 

96614 

.uy 

38636 

.ou 

20 

41 

9.96300 

no 

10.36551  'g 

19 

41 

9.96619 

.09 

AQ 

10.38672 

.60 

AA 

19 

42 
43 

96305 
96311 

.uy 
.09 

•AQ 

36586  1  -J5 
36621  j  'JJ 

18 
17 

42 
43 

96624 
96629 

091   38708 
38744 

.OU 

.60 

CA 

18 
17 

44 
45 

96316 
96322 

.uy 
.09 
no 

36655 
36690 

.f  O 

.58 

16 
15 

44 
45 

96634 
96640 

.uy 
.09 

nn 

38780 
38816 

,OU 

.60 

16 
15 

46 

96327 

.uy 

An 

36725 

CO 

14 

46 

96645 

38852 

14 

47 

96333 

.uy 

Aft 

36760   'J° 

13 

47 

96650 

38888 

fiO   13 

48 

96338 

.uy 
.09 

36795   •:* 

12 

48 

96655 

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38924 

.ou 

fiO 

12 

49 

96343 

36830   •?: 

11 

49 

96660 

.uy 

38960 

.ou 

11 

50 

96349 

.09 

36865   *„ 

10 

50 

96665 

.09 

38996 

.60 

10 

51 

9.96354 

.09 

OQ 

10.  36899  !  '2 

9 

51 

9.96670 

•J  J  j  10.39033 

.60 

An 

9 

52 

96360 

.uy 

36934 

.1^0 

8 

52 

96676 

.uy 

39069 

.OU 

8 

53 

96365 

.09 

36969 

.5£ 

7 

53 

96681 

.09 

39105 

.60 

7 

54 

96370 

.09 

37004   '?* 

6 

54 

96686 

.09 

39141 

.60 

6 

55 
56 

96376 
96381 

.09 
.09 

37039 
37074 

.*5E 

5 
4 

55 
56 

96691 
96696 

.09 
.09 

39177 
39214 

.60 
.60 

5 
4 

57 

96387 

.09 

37110 

.5£ 

3 

57 

96701 

.09 

39250 

.60 

3 

58 

96392 

.09 

37145 

.5£ 

2 

58 

96706 

.09 

39286 

.61 

2 

59 

96397 

.09 

37180 

.Ofc 

1 

59 

96711  ^ 

39323 

.61 

1 

60 

96403 

.09 

37215 

.59 

1 

0 

60 

96717 

.uy 

39359 

.61 

0 

M.  Cosiue. 

Dl" 

Cotang.  1  Dl" 

31. 

31. 

Cosine. 

Dl" 

Cotang. 

Dl" 

M. 

61 


TABLE  IV.— LOGARITHMIC 


69* 


If. 

Siue. 

Dl" 

Tang. 

Dl" 

M. 

M. 

Siue.  |  Dl" 

Taug. 

Dl" 

M. 

0 

9.96717 

10.39359 

60 

0 

9.97015  !n  M 

10.41582 

60 

1 

96722 

O.OJ 

39395 

0.61 
fil 

59 

1 

97020  ,U'n° 

41620 

0.63 

60 

59 

2 
3 
4 

96727 
96732 
96737 

'M 

.08 

39432 

39468 
39505 

.01 
.61 
.61 

58 
57 
56 

2 

3 
4 

97025 
97030 
97035 

!08 

.08 

41658 
41696 
41733 

O 

.63 
.63 

58 
57 
56 

5 

96742 

.Oj 

39541 

.61 

55 

5 

97039 

.08 

41771 

•  vK 

55 

6 

96747 

.Oc 

AC 

39578 

.61 
fil 

54 

6 

97044 

.08 

no 

41809 

.63 

6« 

54 

7 
8 

96752 
96757 

•  Uc 

.08 

AQ 

39614 
39651 

•  O  I 

.61 

fil 

53 
52 

7 
8 

97049 
97054 

.UO 

.08 

no 

41847 

41885 

t 
.63 

6e 

53 
52 

9 

96762 

•  Uc 
no 

39687 

.0  1 
fil 

51 

9 

97059 

.Uo 

AQ 

41923 

£ 
60 

51 

10 

96767 

.Uo 

39724 

.01 

50 

10 

97063 

.Uo 

41961 

f, 

50 

11 

9.96772 

AC 

10.39760 

.61 

A  1 

49 

11 

9.97068 

.08 

AQ 

10.41999 

.62 

49 

12 

96778 

.Uo 

.Of 

39797 

.O  1 

61 

48 

12 

97073 

.Uo 

AQ 

42037 

.62 
63 

48 

13 

96783 

39834 

A  1 

47 

13 

97078 

*UO 

AO 

42075 

47 

14 

96788 

AC 

39870 

.ol 

fil 

46 

14 

97083 

.Uo 

()Q 

42113 

.62 

46 

15 

96793 

•Uc 

no 

39907 

•  Ol 

fil 

45 

15 

•  97087 

•  Uo 

AQ 

42151 

A  /I 

45 

16 

96798 

.Uc 

nc 

39944 

.0  I 
A  1 

44 

16 

97092 

•Uo 

AO 

42190 

.04 

44 

17 

96803 

.Uo 
.08 

39981 

.01 

61 

43 

17 

97097 

•Uo 

AQ 

42228 

.64 

a\ 

43 

18 

96808 

40017 

42 

18 

97102 

•  Uo 

42266 

.0-4 

42 

19 

20 

96813 
96818 

.08 
.08 

nc 

40054 
40091 

.61 

.61 
fi  l 

41 

40 

19 

20 

97107 
97111 

.08 
.08 

42304 
42342 

.64 
.64 

A  1 

41 

40 

21 

9.96823 

.Uc 

10.40128 

.0  L 

39 

21 

9.97116 

•Jf  110.42381 

.O4 

39 

22 

96828 

.08 
08 

40165 

.61 

38 

22 

97121 

08  i   42419 

.64 
fid 

38 

23 

96833 

.Uc 

40201 

.0  I 

37 

23 

97126 

no   42457 

.04 

37 

24 
25 
26 

96838 
96843 

96848 

.08 
.08 
.08 

40238 
40275 
40312 

.62 
.62 
.62 

36 
35 
34 

24 
25 
26 

97130 

97135 
97140 

.08 
.08 

.08 

42496 
42534 
42572 

.64 
.64 
.64 

36 
35 
34 

27 

96853 

.08 
no 

40349 

.62 

33 

27 

97145 

.08 

AC 

42611 

.64 

33 

28 

96858 

.Uo 

.08 

40386 

62 

32 

28 

97149 

•Uo 

AQ 

42649 

.64 

32 

29 

96863 

AQ 

40423 

AO 

31 

29 

97154 

•  Uo 

42688 

/>  t 

31 

30 

96868 

.Uo 

40460 

,02 

30 

30 

97159 

.08 

42726 

.o4 

30 

31 

9.96873 

.08 

AQ 

10.40497 

62 

29 

31 

9.97163 

.08 

AQ 

10.42765 

.64 

29 

32 

96878 

•  UO 

Act 

40534 

AO 

28 

32 

97168 

•  Uo 

AO 

42803 

A  i 

28 

33 

96883 

.Uo 

no 

40571 

.OZ 

27 

33 

97173 

.Uo 

AQ 

42842 

.o4 

RA 

27 

34 

96888 

.Uo 

no 

40609 

62 

26 

34 

97178 

•Uo 

AQ 

42880 

.04 

26 

35 

96893 

.uo 

AQ 

40646 

25 

35 

97182 

•  Uo 
AQ 

42919 

fi4 

25 

36 

96898 

•  UO 

no 

40683 

fi9 

24 

36 

97187 

•  Uo 

AQ 

42958 

.04 

24 

37 

96903 

.Uo 
no 

40720 

.62 

23 

37 

97192 

.Uo 

AQ 

42996 

fi 

23 

38 
39 

96907 
96912 

.uo 
.08 

08 

40757 
40795 

[62 

22 
21 

38 
39 

97196 
97201 

•  Uo 

.08 

AQ 

43035 
43074 

!65 

22 
21 

40 

96917 

.UO 

40832 

•  OZ 

20 

40 

97206 

•  Uo 

43113 

.00 

20 

41 

9.96922 

.08 
no 

10.40869 

.62 
62 

19 

41 

9.97210 

.08 

no 

10.43151 

.65 

AC 

19 

42 

96927 

.uo 

no 

40906 

AO 

18 

42 

97215 

•  Uo 

AQ 

43190 

.00 
Ax 

18 

43 

96932 

.uo 

40944 

.0^ 

17 

43 

97220 

.UO 

43229 

•  DO 

17 

44 

96937 

.08 
no 

40981 

.62 

AO 

16 

44 

97224 

.08 

no 

43268 

.65 

Ax 

16 

45 

96942 

.Uo 

no 

41019 

«OJ5 

15 

45 

97229 

•  UO 

AQ 

43307 

.00 
Ax 

15 

46 

96947 

.uo 

AQ 

41056 

AO 

14 

46 

97234 

.UO 

43346 

•  OO 
AX 

14 

47 

96952 

.Uo 

AQ 

41093 

.oZ 

AO 

13 

47 

97238 

.08 

no 

43385 

.00 

13 

48 
49 

96957 
96962 

.UO 

.08 

AO 

41131 
41168 

•OZ 

.62 

12 
11 

48 
49 

97243 
97248 

.UO 

.08 

AO 

43424 
43463 

!e5 

£  - 

12 
11 

50 
51 
52 

96966 
9.96971 
96976 

.Uo 

.08 
.08 

41206 
10.41243 
41281 

.63 
.63 

10 

9 
8 

50 
51 
52 

97252 
9.97257 
97262 

43502 
'"°  110.43541 

08  i   4358° 

.00 

.65 
.65 

ax 

10 
9 

8 

53 
54 
55 

96981 
96986 
96991 

!08 
.08 

41319 
41356 
41394 

!63 
.63 

7 
6 
5 

53 
54 
55 

97266 
97271 
97276 

43619 

ol  43658 

•S°   43697 

•  DO 

.65 
.65 

A  " 

7 
g 
5 

58 
57 

96996 
97001 

.08 
.08 

41431 
41469 

.63 
.63 

4 

3 

56 
57 

97280  -jjjj   43736 
972851  '55   43776 

.00 

.65 

4 
3 

58 
59 

97005 
97010 

.08 
.08 

41507 
41545 

.63 
.63 

2 

1 

58  |  97289 
59   97294 

.08 

43815 
43854 

.65 
.65 

2 
1 

60 

97015 

.08 

41582 

.63 

0 

60  i  97299'  ' 

43893 

.65 

0 

M. 

Cosine. 

Dl" 

C'ot:in.a. 

Dl" 

.M. 

M.  |  Cosine.   Dl" 

Cotuiig. 

Dl" 

M 

70° 


SINES  AND    TANGENTS. 


M. 

Sine. 

Dl" 

Tang.   Dl' 

M. 

M. 

Sine. 

Dl" 

Taug. 

Dl" 

If. 

0 

9.97299 
97303 

0.08 

10.43893 
43933 

0.66 

60 
59 

0 
1 

9.97567 
97571 

0.07 

]  0.46303 
46344 

0.68 

60 
59 

2 

97308 

.08 

43972 

.66 

58 

2 

97576 

.07 

46385 

.68 

58 

3 

97312 

.08 

44011 

.66 

57 

3 

97580 

.07 

46426 

.69 

57 

4 

97317 

.08 

44051 

.66 

56 

4 

97584 

.07 

46467 

.69 

56 

5 

97322 

.08 

44090 

.66 

55 

5 

97589 

.07 

46508 

.69 

55 

6 

97326 

.08 

44130 

.66 

54 

6 

97593 

•O? 

46550 

.69 

54 

7 

97331 

.08 

44169 

.66 

53 

7 

97597 

.07 

46591 

.69 

53 

8 

97335 

.08 

AQ 

44209 

.66 

52 

8 

97602 

.07 

0-7 

46632 

.69 
-  An 

52 

9 

97340 

.Uo 

44248 

.00 

51 

9 

97606 

7 

46673 

.oy 

51 

10 
11 

97344 

9.97349 

.08 
.08 

44288 
10.44327 

.66 
.66 

50 
49 

10 
11 

97610 
9.97615 

.07 

.07 

46715 
10.46756 

.69 
.69 

50 
49 

12 

97353 

.08 

44367 

.66 

48 

12 

97619 

.07 

46798 

.69 

48 

13 
14 

97358 
97363 

.08 

.08 

44407  '™ 
44446  'JJ 

47 
46 

13 

14 

97623 
97628 

.07 

.07 

46839 
46880 

.69 

.69 

47 
46 

15 

97367 

.08 

444861  '55 

45 

15 

97632 

.07 

46922 

.69 

45 

16 

97372 

.08 

AQ 

44526 

.00 

AA 

44 

16 

97636 

.07 

AT 

46963 

.69 

44 

17 

97376 

.Uo 

44566 

.00 

43 

17 

97640 

.U7 

47005 

.OJ 

43 

18 

97381 

.08 

AQ 

44605 

.66 
/»/> 

42 

18 

97645 

.07 

47047 

.69 

A  A 

42 

19 

97385 

.Mo 

44645 

.On 

41 

19 

97649 

.07 

47088 

.oy 

41 

20 
21 

97390 
9.97394 

.08 

.08 

AQ 

44685 
10.44725 

.66 
.66 

AT 

40 
39 

20 
21 

97653 
9.97657 

.07 
.07 

47130 
10.47171 

.69 
.69 

40 
39^ 

22 

97399 

.Uo 

44765 

.Of 

38 

22 

97662 

.07 

47213 

.70 

38 

23 

97403 

.08 

44805 

.67 

37 

23 

97666 

.07 

47255 

./O 

37 

24 

25 

97408 
97412 

.07 
.07 

44845 

44885 

.67 
.67 

31) 
35 

24 
25 

97670 
97674 

.07 
.07 

47297 
47339 

.70 
.70 

36 
35 

26 

97417 

.07 

07 

44925 

.67 

67 

34 

26 

97679 

.07 

47380 

.70 

tf  A 

34 

27 

97421 

•  U  t 

44965 

I 

33 

27 

97683 

.07 

47422 

.  1  U 

33 

28 

97426 

.07 

45005 

.67 

32 

28 

97687 

.07 

47464 

.70 

32 

29 

97430 

.07 

45045 

.67 

6^ 

31 

29 

97691 

.07 

A*7 

47506 

.70 

31 

30 

97435 

.07 

45085 

i 

30 

30 

97696 

.07 

47548 

.70 

30 

31 

9.97439 

.07 

10.45125 

.67 

29 

31 

9.97700 

.07 

10.47590 

.70 

29 

32 
33 

97444 
97448 

.07 
.07 

AT 

45165 
45206 

.67 
.67 

28 
27 

32 
33 

97704 
97708 

.07 

.07 

0— 

47632 
47674 

.70 
.70 

28 
27 

34 

97453 

•  VI 

45246 

•"' 

26 

34 

97713 

t 

47716 

.70 

26 

35 

97457 

.07 

45286 

.67 

25 

35 

97717 

.07 

47758 

.70 

25 

36 
37 

97461 

97466 

.07 
.07 

45327  '"' 
45367  '!; 

24 
23 

36 
37 

97721 
97725 

.07 
.07 

47800 
47843 

.70 
.70 

24 
23 

38 

97470 

.07 

07 

45407  'r' 

22 

38 

97729 

07 

47885 

.70 

70 

22 

39 

97475 

•  "  / 

45448  'H 

21 

39 

97734 

.VI 

47927 

.  l() 

21 

40 

97479 

07 

45488  ™J 

20 

40 

97738 

.07 

/IT 

47969 

.70 

20 

41 

9.97484 

•  U  t 

10.45529  rJJ 

19 

41 

9.97742 

.U  I 

10.48012 

.71 

19 

42 

97488 

.07 

455691  'JJ 

18 

42 

97746 

.07 

48054 

.71 

18 

43 

97492 

.07 

45610!  -Jo 

17 

43 

97750 

.07 

48097 

.71 

17 

44 

97497 

.07 

456501  ••„ 

16 

44 

97754 

.07 

48139 

.7] 

16 

45 

97501 

.07 

AT 

45691 

.00 

15 

45 

97759 

.07 

48181 

.71 

15 

46 

97506 

Ml 

AT 

45731 

AQ 

14 

46 

97763 

.07 

07 

48224 

.71 

71 

14 

47 

97510 

.11  4 

AT 

45772 

•  Oo 

13 

47 

97767 

.U  t 

48266 

.«  1 

13 

48 

97515 

.0* 
AT 

45813 

.68 

12 

48 

97771 

.07 

48309 

17  1 

12 

49 

97519 

.U  1 

45853 

.68 

11 

49 

97775 

.07 

48352 

.71 

11 

50 

97523 

.07 

07 

45894 

.68 

AQ 

10 

50 

97779 

•J.    48394 

.71 

7"1 

10 

51 

9.97528 

•  U  t 

10.45935 

.Oo 

9 

51 

9.97784 

'\  10.48437 

•  1  i 

9 

52 

97532 

.07 

AT 

45975 

.68 

8 

52 

97788 

'AT    48480 

.71 

8 

53 

97536 

.0  t 
AT 

46016 

.68 

AQ 

7 

53 

97792 

.07 

0  — 

48522 

.71 

7 

54 
55 

97541 
97545 

.U  t 

.07 

46057 
46098 

.Oo 
.68 

6 
5 

54 

55 

97796 
97800 

t 

.07 

48565 
48608 

.71 
.71 

6 
5 

56 
57 

97550 
97554 

.07 
.07 

46139 
46180 

.68 
.68 

4 
3 

56 
57 

97804 
97808 

.07 
.07 

48651 
48694 

.71 

.71 

4 
3 

58 

97558 

.07 

46221 

.68 

2 

58 

97812 

.07 

48736 

.72 

2 

59 

97563 

.07 

07 

46262  '™ 

1 

59 

97817 

.07 

ft  7 

48779 

.72 

1 

60 

97567 

*U  ( 

46303  ' 

0 

60 

97821 

•  U  t 

48822 

•  ^ 

0 

31. 

Cosine. 

Dl" 

Cotang.  I  Dl" 

31 

M. 

Cosine. 

Dl" 

Cotang. 

Dl" 

~M7 

19C 


63 


72° 


TABLE  IV.— LOGARITHMIC 


M. 

Sine. 

D." 

Tang. 

1)1" 

M. 

31.    Sine. 

Dl" 

Tan*. 

Di" 

31. 

0 

1 

9.97821 
97825 

0.07 

10.48822 

48865 

0.72 

60 
59 

0 

9.98060 
98063 

0.06 

10.51466 
51511 

0.75 

60 
59 

2 

97829 

.07 

48908 

.72 

58 

2 

98067!  -™ 

51557 

.75 

58 

4 

5 

97833 
97837 
97841 

.07 
.07 
.07 

48952 
48995 
49038 

.72 
.72 
.72 

57 
56 
55 

3 
4 
5 

98071 
98075 
98079 

.ub 
.06 
.06 

51602 
51647 
51693 

.75 
.76 
.76 

57 
56 
55 

6 

97845 

.07 
.07 

4906! 

.72 

54 

6 

98083 

.06 
OH 

5  1738 

.76 

54 

7 

97849 

49124 

.•  ^ 

53 

7 

98087 

.UO 

51783 

•' 

53 

8 

97S53 

.07 
.07 

49167 

.72 

7.) 

52 

8 

98090 

.06 
or. 

51829 

.76 
7fi 

52 

9 

10 

97857 
97861 

.*07 

07 

49211 
49254 

.  I  Z 

.72 

TO 

51 

50 

9 
10 

98094 
98098 

•UO 

.06 

AC 

51874 
51920 

.  i  0 

.76 

ft  G 

51 

50 

11 

9.97866 

/ 

.07 

10.49297 

./  L 

79 

49 

11 

9.98102 

•Do 

M, 

10.51965 

.7o 

49 

12 

97870 

49341 

.  i  Z 

48 

12 

98106 

•  UO 

52011 

.10 

48 

13 

97874 

.07 

(17 

49384 

.72 

7O 

47 

IS 

981  10 

.06 

52057 

.76 

7  A 

47 

14 

97878 

.0  t 

49428 

,t  L 

46 

14 

98113 

.06 

52103 

.  /  0 

46 

15 

97882 

.07 

07 

49471 

.72 

45 

15 

98117 

.06 

AA 

52148 

.76 

7fi 

45 

16 

97886 

.U  i 

49515 

,73 

44 

16 

98121 

.UO 

52194 

.  *  0 

44 

17 

97890 

.07 
.07 

49558  •£ 

43 

17 

98125 

.06 

52240 

.76 

76 

43 

18 

97894 

.07 

49602 

73 

42 

18 

98129 

OH 

52286 

•  i  O 

.77 

42 

19 

97898 

07 

49645 

41 

19 

98132 

•  UO 
(\K 

52332 

41 

20 

97902 

.U  t 

49689 

.'•» 

40 

20 

98136 

•  UO 

52378 

'l- 

40 

21 

9.97906 

.07 
.07 

10.49733 

.73 
73 

39 

21 

9.98140 

.06 
06 

10.52424 

.''  i 

39 

22 

97910 

49777 

38 

22 

98144 

.UO 

52470 

' 

38 

2.5 

97914 

.07 
07 

49820 

.73 

37 

23 

98147 

.06 
06 

52516 

.77 

37 

24 

97918 

•  U  i 

49864 

.'  *• 

36 

24 

98151 

52562 

.  it 

36 

25 

97922 

.07 

49908 

.73 

35 

25 

98155 

.06 

52608 

.77 

35 

26 

97926 

.07 

49952 

.73 

34 

26 

98159 

.06 

52654 

.77 

34 

27 

979:50 

.07 

49996 

.73 

S3 

27 

98162 

.06 

A  A 

52701 

.77 

77 

33 

28 
29 

97934 
97938 

.*07 

07 

50040 
50084 

!73 

32 
31 

28 
29 

98166 
98170 

.UO 

.06 

n/» 

52747 
52793 

.  I  ' 

.77 

77 

32 
31 

30 
31 

97942 
9.97946 

i 

.07 

50128 
10.50172 

'.?* 

30 
29 

30 
31 

98174 
9.98177 

•Do 

.06 

52840 
10.52886 

t 

.77 

30 
29 

32 

97950 

.07 

A7 

50216 

.74 

28 

32 

98181 

.06 

AA 

52932 

.77 

77 

28 

33 

97954 

.0  1 

50260 

.74 

27 

33 

98185 

.lit) 

52979 

t 

27 

35 
36 
37 

97958 
97962 
97966 
97970 

.07 
.07 
.07 
.07 

50304 
50348 
5039:5 
50437 

.74 
.74 
.74 
.74 

26 
25 
24 
23 

34 
35 
36 
37 

98189 
98192 
98196 
98200 

.06 
.06 
.06 
.06 

53025 
53072 
53U9 
53165 

'.78 
.78 
.78 

26 
25 
24 
23 

38 

97974 

.07 

50481 

.74 

22 

38 

98204  •:::: 

53212 

•'" 

22 

39 

97978 

.07 

50526 

.74 

21 

39 

98207 

.UO 

53259 

.78 

21 

40 
41 

97982 
9.97986 

.07 
.07 

50570 
10.50615 

.74 
.74 

20 
19 

40 
41 

98211 
9.98215 

.06 
.06 

53306 
10.53352 

!78 

20 
19 

42 

97989 

.07 

50659 

.74 

18 

42 

98218 

.06 

53399 

.to 

18 

43 

97993 

.07 

50704 

.74 

17 

43 

98222  •"" 

53446 

17 

44 

97997 

.07 

50748 

.74 

16 

44 

98226  '}!!! 

53493 

.78 

16 

45 
46 

98001 
98005 

.07 
.07 

50793 
50837 

.74 
.74 

15 
14 

45 
46 

98229 
98233 

.UO 

.06 

63540 
63587 

.78 

.78 

15 
14 

47 

98009 

.07 

50882 

.74 

13 

47 

98237  •;;;; 

53634 

.7M 

13 

48 

98013 

.07 

50927 

.75 

12 

48 

98240  '  £ 

53681 

.79 

12 

49 

98017 

.07 

50971   'I.' 

11 

49 

98244  •  J 

53729 

.79 

11 

50 

98021 

.07 

51016 

.<  j 

10 

50 

98248,  •",! 

53776 

.79 

10 

51 

9.98025 

.07 

10.51061 

.75 

9 

51 

9.98251 

.UO 

t\t\ 

10.53823 

.79 
70 

9 

52 

98029 

.06 

51106 

.  t  ,> 

8 

52 

98255 

•  UO 

53870 

.  i  y 

8 

53 

98032 

.1)6 

51151 

.75 

7 

53 

98259 

.06 

53918 

.79 

~  »  t 

7 

54 

98036 

.06 

51196 

.75 

6 

54 

98262 

.06 

53965 

.79 

6 

55 

98M40 

.06 

51241 

.75 

5 

55 

98266 

.06 

54013 

.79 

5 

56 

98044 

.06 

51286 

.75 

4 

56 

98270 

.06 

54060 

.79 

4 

57 

98048 

.06 

51331 

.75 

3 

57 

98273 

.06 

54108 

"I 

3 

58 

98052 

.06 

51376 

.75 

2 

58 

98277 

.06 

54155 

.  t  y 

rrf\ 

2 

59 

98056 

.06 

51421 

.75 

1 

59 

98281 

.06 

5420H 

.79 
it\ 

1 

0  U 

98060 

.06 

51461 

.75 

0 

60 

98284 

.06 

54250 

.  <  y 

0 

M. 

Cosine.   HI" 

Cotang. 

w> 

M. 

M. 

Cosine. 

~D~i" 

Cotang 

"DT 

M. 

17° 


SINES  AND  TANGENTS. 


75C 


M. 

Sine.  I  Dl" 

Tang.  I  Dl" 

M. 

M.  |-  Sine.   Dl" 

Tanx. 

Dl" 

M. 

0 
1 

J.98284 

98288 

0.06 
A(> 

10,54250 
54298 

0.80 

60 
59 

0  1 
1 

9.98494 

98498 

0.06 
06 

10.57195 
57245 

0.84 

fiO 
59 

2 
3 

4 
5 

98291 
98295 
98299 
98302 

.uo 
.06 
.06 
.06 

54346 
54394 
54441 
54489 

.80 
.80 
.80 

OA 

58 
57 
56 
55 

2 
3 
4 
5 

98501 
98505 
98508 
98511 

.*06 
.06 
.06 

Ofi 

57296 
57347 
57397 

57448 

M 
.85 
.85 
85 

58 
57 
56 
55 

6 

7 

98306 
98309 

'.06 

54537 
54585 

•  oU 

.80 

54 
53 

6 

7 

98515 
98518 

.uo 
.06 

57499 
57550 

!85 

QC 

54 

53 

8 

98313 

.06 

54633 

.80 

52 

8 

98521 

.06 

AC 

57601 

.00 

52 

9 
10 

98317 
98320 

!o6 

54681 
54729 

'.80 

51 

50 

9 
10 

98525 

98528 

.uo 

.06 

57652 
57703 

!85 

OC 

51 

50 

11 

9.98324 

.06 

10.54778 

.80 

P.O 

49 

It 

9.98531 

06 

10.57754 

.80 

85 

49 

12 

98327 

.Uo 

54826 

•  8U 

48 

12 

98535 

.Uv) 

57805 

48 

13 

98331 

.06 

AC 

54874 

.80 

47 

13 

98538 

.06 
06 

57856 

85 

47 

14 

98334 

.UO 

54922 

.80 

46 

14 

98541 

•  UO 

57907 

46 

15 

98338 

.06 

54971 

.81 

0  1 

45 

15 

98545 

06 

57959 

*86 

45 

16 
17 

98342 
98345 

!06 

55019 
55067 

.0  I 

.81 

-   0  1 

44 
43 

16 
17 

98548 
98551 

•  UO 

.06 
ofi 

58010 

58061 

!86 
86 

44 
43 

18 
19 

98349 
98352 

.06 

55116 
55164 

•  O  I 

.81 

42 
41 

18 
19 

98555 
98558 

•  UO 

.06 

AC 

58113 
58164 

[86 

CO 

42 
41 

20 

98356 

.06 

Ofi 

55213 

.81 

Q  I 

40 

20 

98561 

.Uo 

nfi 

58216 

.80 

86 

4« 

21 

9.1)8359 

•Uo 

10.55262 

.8  I 

39 

21 

9.98565 

•  UO 

10.58267 

QC 

39 

22 

98363 

.06 

55310 

.81 

Q  1 

38 

22 

98568 

.06 

A  t 

58319 

.80 

38 

23 

98366 

.uo 

~  55359 

.8  1 

Q  1 

37 

23 

98571 

.UO 
AR 

58371 

*86 

37 

24 

98370 

.06 

55408 

.81 

36 

24 

98574 

.UO 

58422 

36 

25 

98373 

.06 

55456 

.81 

Q  1 

35 

25 

98578 

.05 

At 

58474 

86 

35 

26 
27 

98377 
98381 

'.06 
Ofi 

55505 
55554 

.01 

.81 

34 

33 

26 

27 

98581 
98584 

.UO 

.05 

.05 

58526 

58578 

.00 

.87 

87 

34 
33 

28 

98384 

.uo 

55603 

•**• 

32 

28 

98588 

58630 

OT 

32 

29 

98388 

.06 

Ofi 

55652 

.82 

31 

29 

98591 

.05 
.05 

58682 

.87 
87 

31 

30 
31 
32 
33 
34 

98391 
9.98395 
98398 
98402 
98405 

.uo 
.06 
.06 
.06 
.06 

55701 
10.55750 
55799 
55849 

55898 

!82 
.82 
.82 
.82 

QO 

30 
29 
28 
27 
26 

30 
31 
32 
33 
34 

98594 
9.98597 
98601 
98604 
98607 

.05 
.05 
.05 
.05 

58734 
10.58786 
58839 
58891 
58943 

'.87 
.87 
.87 
.87 
07 

30 
29 
28 
27 
26 

35 

98409 

Ofi 

55947 

.06 
OO 

25 

35 

98610 

.05 

58995 

.01 

25 

36 

98412 

.uo 

Ofi 

55996  '™ 

24 

36 

98614 

.05 

59048 

.87 

24 

37 

98415 

.uo 

56046 

.V* 

23 

37 

98617 

t\~ 

59100 

DO 

23 

38 
39 

98419 
98422 

.1)6 

56095 
56145 

.82 
.82 

22 
21 

38 
39 

98620 
98623 

.Uo 
.05 

-  59153 
59205 

.88 

.88 

QO 

22 
21 

40 

98426 

.06 

Ofi 

56194  •£) 

20 

40 

98627 

•*? 

59258 

.OO 

QQ 

20 

41 
42 
43 

9.98429 
98433 
98436 

•  UO 

.06 
.06 

10.56244 
56293 
56343 

.00 
.83 
.83 

19 
18 
11 

41 

42 
43 

9.98630 
98633 
98636 

•J-J  10.59311 
f\   59364 
•J?    59416 

.00 

.88 
.88 

QQ 

19 
18 
17 

44 

9844oi  -;;° 

56393  •*? 

16 

44 

98640 

•J?    59469 

.80 

16 

45 

98443  i  ' 

56442 

.00 

15 

45 

98643 

59522 

.88 

UCJ 

15 

46 

98447  '  « 

56492 

.83 

14 

46 

98646 

At 

59575 

.88 

00 

14 

47 

48 

98450;  '  « 
98453  ' 

56542'  '^ 
56592  rj» 

13 
12 

47 
48 

98649 
98652 

.UO 

.05 

59628 
59681 

.88 

.88 

w  (  . 

13 
12 

49 

98457 

.uo 

Ofi 

56642  i  •?:: 

11 

49 

98656 

.05 

59734 

.ay 

on 

11 

50 

98460 

.uo 

Oft 

56692  'JJ 

10 

50 

98659 

A?! 

597881  -^ 

10 

51 

9.98464 

•  IJt 

10.56742  •*' 

9 

51 

9.98662 

•JJ  10.59841 

Q{\ 

9 

52 

98467 

.06 

56792  '°* 

8 

52 

98665 

•II   .  59894 

.sy 

8 

53 

9847 

.Oe 

56842  •**} 

7 

53 

98668 

•"?    59948 

.89 

7 

54 

9S474 

.06 

56892  i  ~1 

6 

54 

98671 

•'?    60001 

.8.9 

Oft 

6 

55 

98477 

.06 

56943  •*} 

5 

55 

9S675 

•~f.    60055 

.8iJ 

5 

56 

98481 

.06 

56993  ~j 

4 

56 

98678 

•  J    60108 

,8S 

4 

57 

98484 

.Of 

AC 

57043  !  -°] 

3 

57 

98681 

>Jr    60162 

.8^ 

3 

58 

98488 

.Uo 

57094  ~1 

2 

58 

98684 

•J?    60215 

.8^ 

2 

59 

98491 

.Ut 

57144  *". 

1 

59 

98687 

•JJ    60269 

iVt 

1 

60 

98494 

.06 

57195 

.84 

0 

60 

98690 

fi0323 

.9( 

0 

.M. 

Cosine. 

Dl"   Cotmig. 

Dl" 

M. 

M. 

Cosiiu'. 

Dl"   Cotang. 

Dl" 

M. 

13C 


76C 


TABLE  IV.— LOGARITHMIC 


77° 


M. 

Sine. 

Dl" 

Tang. 

Di': 

M. 

M. 

Sine. 

Dl" 

Tang. 

Dl" 

M. 

0 

1 

9.98690 
98694 

0.05 

10.60323 
60377 

0.90 

60 
59 

0 
1 

9.98872 
98875 

0.05 

A' 

10.63664 
63721 

0.96 

60 

59 

2 

98697 

.05 

60431 

.90 

58 

2 

98878 

.Oo 

63779 

.96 

58 

3 

98700 

.05 

60485 

.90 

57 

3 

98881 

.05 

63837 

.96 

57 

4 

98703 

.05 

60539 

.90 

56 

4 

98884 

.05 

63895 

.96 

56 

5 

98706 

.05 

60593 

.90 

55 

5 

98887 

.05 

63953 

.97 

55 

6 

98709 

.05 

60647 

.90 

54 

6 

9S890 

.05 

64011 

.97 

54 

7 

98712 

.05 

AX 

60701 

.90 

/\A 

53 

7 

98893 

.05 

Ar 

64069 

.97 

53 

8 
9 
10 

98715 
98719 
98722 

.Uo 
.05 
.05 

60755 
60810 
60864 

.yu 
.91 
.91 

52 
51 
50 

8 
9 
10 

98896 
98898 
98901 

I.Oo 
.05 
.05 

64127 
64185 
64243 

.97 
.97 
.97 

52 
51 
50 

11 
12 

9.98725 

98728 

.05 
.05 

10.60918 
60973 

.91 
.91 

49 

48 

11 

12 

9.98904 
98907 

.05 
.05 

10.64302 
64360 

.97 

'11 

49 

48 

13 
14 

98731 
98734 

.05 
.05 

61028 
61082 

.91 
.91 

47 

46 

13 
14 

98910 
98913 

.05 
.05 

64419 
64477 

,98 
.98 

47 
46 

15 

98737 

.05 

61137 

.91 

45 

15 

98916 

.05 

64536 

.98 

45 

16 

98740 

.05 

A  •" 

61192 

.91 

44 

16 

98919 

.05 

64595 

.98 

44 

17 

98743 

.Uo 

61246 

.91 

43 

17 

98921 

.05 

64653 

.98 

43 

18 

98746 

.05 

A  c 

61301 

.91 

42 

18 

98924 

.05 

64712 

.98 

42 

19 

98750 

.U5 

61356 

.92 

41 

19 

98927 

.05 

64771 

.98 

41 

20 

98753 

.05 

61411 

.92 

40 

20 

98930 

.05 

64830 

.98 

40 

21 
22 

9.98756 
98759 

.05 
.05 

A  - 

10.61466 
61521 

.92 
.92 

no 

39 
38 

21 
22 

9.98933 
98936 

.05 
.05 

A  - 

10.64889 
64949 

.98 
.99 

39 

38 

23 
24 
25 

98762 
98765 

98768 

»UO 

.05 
.05 

61577 
61632 
61687 

.»2 
.92 
.92 

37 
36 
35 

23 
24 
25 

98938 
98941 
98944 

.1)0 

.05 
.05 

65008 
65067 
65126 

.99 
.99 
.99 

37 

36 
35 

26 

98771 

.05 

61743 

.92 

34 

26 

98947 

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65186 

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34 

27 

98774 

.05 

61798 

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33 

27 

98950 

.05 

65245 

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33 

28 

98777 

.05 

61853 

.93 

32 

28 

98953 

.05 

65305 

.99 

32 

29 

98780 

.05 

61909 

.93 

31 

29 

98955 

.05 

65365 

.99 

31 

30 

98783 

.05 

61965 

.93 

30 

30 

98958 

.05 

65424 

.00 

30 

31 

9.98786 

.05 

Ar 

10.62020 

.93 

no 

29 

31 

9.98961 

.05 

A  r. 

10.65484 

.00 

29 

32 
33 

98789 
98792 

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.05 

AX 

62076 
62132 

.\?6 

.93 

QO 

28 
27 

32 

33 

98964 
98967 

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.05 

A  P. 

65544 
65604 

.00 
.00 

28 
27 

34 

98795 

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62188 

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26 

34 

98969 

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65664 

.00 

26 

35 

98798 

.05 

Ar 

62244 

.93 

25 

35 

98972 

.05 

65724 

.00 

25 

36 

98801 

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62300 

.93 

24 

36 

98975 

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65785 

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24 

37 

98S04 

.05 
A- 

62356 

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o  4 

23 

37 

98978 

.05 

A  C 

65845 

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A1 

23 

38 

98807 

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A  H 

62412 

.»4 

22 

38 

98980 

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65905 

•  Ul 

22 

39 

98810 

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A  Z. 

62468 

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f  1  4 

21 

39 

98983 

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A  P. 

65966 

.01 

A1 

21 

40  ' 

98813 

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A  " 

62524 

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20 

40 

98986 

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66026 

.01 

20 

41 

9.98816 

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10.62581 

.94 

19 

41 

9.98989 

.05 

10.66087 

.01 

19 

42 

98819 

.05 

62637 

.94 

18 

42 

98991 

.05 

66147 

.01 

18 

43 
44 

98822 
98825 

.05 
.05 

62694 
62750 

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.94 

17 
16 

43 
44 

98994 
98997 

.05 
.05 

66208 
66269 

.01 
.01 

17 
16 

45 

98828 

.05 

0- 

62807 

.94 

CIA 

15 

45 

99000 

.05 

AT 

66330 

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AO 

15 

46 
47 

98831 
98834 

j 
.05 

A  » 

62863 
62920 

.»4 
.95 

14 

13 

46 
47 

99002 
99005 

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.05 

f\- 

66391 
66452 

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.02 

AO 

14 
13 

48 

98837 

.1)0 

AC 

62977 

.95 

Ar 

12 

48 

99008 

•  Uo 

A  C 

66513 

.02 

no 

12 

49 
50 
51 
52 

98840 
98843 
9.98846 
98849 

.MO 
.05 
.05 
.05 

63034 
63091 
10.63148 
63205 

.yo 
.95 
.95 
.95 

11 
10 
9 

8 

49 
50 
51 
52 

90011 
99013 
9.99016 
99019 

*UD 

.05 
.05 
.05 

66574 
66635 
10.66697 
66758 

.02 

.02 
.02 

.02 

11 

10 
9 

8 

53 

98852 

.05 

A  - 

63262 

.95 

f\  - 

7 

53 

99022 

.05 

A  X 

66820 

.03 

AO 

7 

54 
55 

98855 
98858 

.Uo 
.05 

63319 
63376 

.yo 
.95 

6 
5 

54 
55 

99024 
99027 

.UO 

.05 

66881 
60943 

.Uo 

1.03 

6 
5 

56 

98861 

.05 

A  £ 

63434 

.96 

4 

56 

99030 

.05 

A  - 

67005 

.03 

AO 

4 

'57 

98864 

.UO 

63491 

3 

57 

99032 

.UO 

67067 

•  Uo 

3 

58 

98867 

.05 

63548 

.96 

2 

58 

99035 

.04 

67128 

.03 

2 

59 
60 

98869 
98872 

.05 
.05 

63606 
63G64 

.96 
.96 

1 

0 

59 
60 

99038 
99040 

.04 
.04 

67190 
67253 

1.03 
1.03 

1 

0 

M. 

Cosine. 

Dl" 

Cotans?. 

Dl" 

M. 

.M. 

Cosine-. 

Dl" 

Cotanjr. 

Dl" 

II. 

13° 


G6 


12° 


78° 


SINES  AND    TANGENTS. 


79° 


31. 

Siue. 

Dl" 

Tang. 

Dl" 

31. 

31. 

Sine. 

Dl" 

Tang. 

Dl" 

31. 

0 
1 

9.99040 
99043 

0.04 

10.67253 
67315 

1.04 

60 
59 

0 
1 

9.99195 
99197 

0.04 

10.71135 

71202 

.13 

60 
59 

2 

99046 

.04 

67377 

1.04 

58 

2 

99200 

.04 

71270 

.13 

58 

3 

99048 

.04 

67439 

1.04 

57 

3 

99202 

.04 

71338 

.13 

57 

4 

99051 

.04 

67502 

1.04 

56 

4 

99204 

.04 

71405 

1.13 

56 

5 

99054 

.04 

67564 

1.04 

55 

5 

99207 

.04 

71473 

1.13 

55 

6 

99056 

.04 

67627 

1.04 

54 

6 

99209 

.04 

71541 

1.13 

54 

7 

99059 

.04 

67689 

1.04 

53 

7 

99212 

.04 

71609 

1.13 

53 

8 

99062 

•  04 

67752 

1.05 

52 

8 

99214 

.04 

71677 

1.14 

52 

9 

99064 

.04 

67815 

.05 

A  £ 

51 

9 

99217 

.04 

A  1 

71746 

1.14 
11  \ 

51 

10 

99067 

.04 

67878 

.00 

50 

10 

99219 

.04 

71814 

.14 

50 

11 

9.99070 

.04 

10.67941 

.05 

49 

11 

9.99221 

.04 

10.71883 

1.14 

49 

12 

99072 

.04 

68004 

.05 

48 

12 

99224 

.04 

71951 

1.14 

48 

13 

99075 

.04 

68067 

.05 

47 

13 

99226 

.04 

72020 

1.14 

47 

14 

99078 

.04 

68130 

.05 

46 

14 

99229 

.04 

72089 

1.15 

46 

15 

99080 

.04 

68194 

.06 

45 

15 

99231 

.04 

72158 

1.15 

45 

16 

99083 

.04 

68257 

.06 

44 

16 

99233 

.04 

72227 

1.15 

44 

17 

99086 

.04 

68321 

.06 

43 

17 

99236 

.04 

72296 

1.15 

43 

18 

99088 

.04 

68384 

1.06 

42 

18 

99238 

.04 

72365 

1.15 

42 

19 

99091 

.04 

68448 

1.06 

41 

19 

99241 

.04 

72434 

1.16 

41 

20 
21 
22 

99093 
9.99096 
99099 

.04 
.04 

.04 

68511 
10.68575 
68639 

1.06 
1.06 
1.07 

40 
39 

38 

20 
21 
22 

99243 
9.99245 

99248 

.04 
.04 
.04 

A  A 

72504 
10.72573 
72643 

1.16 
1.16 
1/16 

40 
39 

38 

23 

99101 

.04 

68703 

.07 

37 

23 

99250 

.04 

72712 

1.16 

37 

24 
25 

99104 
99106 

.04 
.04 

68767 
68832 

1.07 
1.07 

1A>7 

36 
35 

24 
25 

99252 
99255 

.04 
.04 

A/I 

72782 
72852 

1.16 
1.17 

36 
35 

26 

99109 

.04 

68896 

.07 

34 

26 

99257 

.04 

72922 

.17 

34 

27 

99112 

.04 

68960 

1.07 

33 

27 

99260 

.04 

A/I 

72992 

1.17 

33 

28 

99114 

•  04 

69025 

1.07 

1  AQ 

32 

28 

99262 

.04 

A  f 

73063 

1.17 

11  *7 

32 

29 

99117 

.04 

69089 

.08 

31 

29 

99264 

.04 

73133 

.17 

31 

30 
31 

32 
33 
."54 

99119 
9.99122 
99124 
99127 
99130 

.04 
.04 
.04 
.04 
.04 

69154 
10.69218 
69283 
69348 
69413 

1.08 
1.08 
1.08 
1.08 
1.08 

30 
29 
28 
27 
26 

30 
31 
32 
33 
34 

99267 
9.99269 
99271 
99274 
99276 

.04 
.04 
.04 
.04 
.04 

73203 
10.73274 
73345 
73415 

73486 

1.17 
1.18 
1.18 
1.18 
1.18 

30 
29 

28 
27 
26 

35 

99132 

.04 

69478 

1  .08 

25 

35 

99278 

.04 

73557 

1.18 

25 

36 
37 
38 
39 

99135 
99137 
99140 
99142 

•04 
.04 
.04 
.04 

69543 
69609 
69674 
69739 

1.09 
1.09 
1.09 
1.09 

24 
23 
22 
21 

36 
37 
38 
39 

99281 
99283 
99285 
99288 

.04 
.04 
.04 
.04 

73628 
73699 
73771 

73842 

1.19 
1.19 
1.19 
1.19 

24 
23 
22 
21 

40 
41 
42 
43 
44 

99145 
9.99147 
99150 
99152 
99155 

.04 
.04 
.04 
.04 
.04 

A  /I 

69805 
10.69870 
69936 
70002 
70068 

1.09 
1.09 
1.10 
1.10 
1.10 
11  1\ 

20 
19 
18 
17 
16 

40 
41 
42 
43 
44 

99290 
9.99292 
99294 
99297 
99299 

•.04 
.04 
.04 
.04 
.04 

A  1 

73914 
10.73985 
74057 
74129 
74201 

1.19 
1.19 
1.20 
1.20 
1.20 
1*iii 

20 
19 
18 
17 
16 

45 

99157 

•  04 

70134 

.10 

15 

45 

99301 

.04 

74273 

.20 

15 

46 

99160 

.04 

A  1 

70200 

1.10 

14 

46 

99304 

.04 

A  1 

74345 

1.20 
101 

14 

47 

99162 

•  04 

70266 

1  A 

13 

47 

&9306 

.04 

74418 

.21 

13 

48 

99165 

.04 

70332 

.10 

12 

48 

99308 

.04 

74490 

1.21 

12 

49 

99167 

.04 

70399 

.11 

11 

49 

99310 

.04 

74563 

1.21 

11 

50 

99170 

.04 

A/l 

70465 

.1  ] 

1  -1 

10 

50 

99313 

.04 

A/I 

74635 

1.21 
101 

10 

51 

9.99172 

.04 

10.70532 

.1  1 

9 

51 

9.99315 

.04 

10.74708 

.21 

9 

52 

99175 

.04 
f>4 

70598 

.11 

8 

52 

99317 

04 

74781 

1.21 

199 

8 

53 

99177 

•  U4 
A/I 

70665 

1  1 

7 

53 

99319 

.04 

A  ,1 

74854 

..2.6 

7 

54 

99180 

.04 

.04 

70732 

.  1  1 

.12 

6 

54 

99322 

.04 

04 

74927 

1  99 

6 

55 

99182 

70799 

5 

55 

99324 

•  U-i 

75000 

1.2£ 

5 

56 
57 

99185 
99187 

.04 

70866 
70933 

.12 
.12 

4 
3 

56 
57 

99326 
9932r: 

.04 
.04 

75074 
75147 

1.22 
1.22 

4 
3 

58 

99190 

.04 

A/I 

71000 

.12 
11  •> 

2 

58 

99331 

.04 

A/I 

75221 

1.23 

100 

2 

59 
60 

99192 
99195 

.04 

.04 

71067 
71135 

.1  L 

L12 

1 

0 

59 
60 

99333 
99335 

.04 
.04 

75294 
75368 

»20 

1.23 

1 

0 

M. 

Cosine. 

Dl" 

Cotang. 

Dl" 

M. 

M. 

Cosine. 

Dl" 

Cotang. 

Dl" 

M. 

11° 


67 


10° 


80° 


TABLE  IV.— LOGARITHMIC 


81° 


M. 

Sine. 

Dl" 

Tang. 

Dl" 

M. 

M. 

Sine. 

Dl" 

Tang. 

Dl" 

M. 

0 
1 

9.99335 
99337 

0.04 

10.75368 
75442 

1.23 

60 
59 

0 
1 

9.99462 
99464 

0.03 

10.80029 
80111 

1.36 

60 
59 

2 

•  99340 

.04 

75516 

.23 

58 

2 

99466 

.03 

80193 

1.37 

58 

3 

99342 

.04 

75590 

.24 

57 

3 

994G8 

.03 

80275 

1.37 

57 

4 

99344 

.04 

75665 

.24 

56 

4 

99470 

.03 

80357 

1.37 

56 

5 

99346 

.04 

n  4 

75739 

.24 

55 

5 

99472 

.03 

no 

80439 

1.37 

1.JQ 

55 

6 

99348 

.04 

75814 

.24 

54 

6 

99474 

.Uo 

80522 

.OO 

54 

7 
8 

99351 
99353 

.04 
.04 

75888 
75963 

.24 
.25 

53 
52 

7 
8 

99476 

99478 

.03 
.03 

80605 
80688 

1.38 
1.38 

53 
52 

9 

99355 

.04 

(\A 

76038 

.25 

o  ^ 

51 

9 

99480 

.03 

no 

80771 

1.38 

51 

10 

99357 

.04 

76113 

.20 

50 

10 

99482 

.Uo 

80854 

1  .39 

50 

11 

9.99359 

.04 

10.76188 

.25 

49 

11 

9.99484 

.03 

10.80937 

1.39 

49 

12 

99362 

.04 

f\4 

76263 

.25 

48 

12 

99486 

.03 

MO 

81021 

1.39 

Ion 

48 

13 
14 
15 

99364 
99366 
99368 

.04 

.04 
.04 

76339 
76414 
76490 

!26 
.26 

47 
46 
45 

13 
14 
15 

99488 
99490 
99492 

.Uo 

.03 
.03 

81104 

81188 
81272 

.oy 

1.40 
1.40 

47 
46 
45 

16 

99370 

.04 

76565 

1.26 

44 

16 

99494 

.03 

81356 

1.40 

44 

17 

18 

99372 
99375 

.04 
.04 

76641 
76717 

1.26 
1.27 

43 
42 

17 

18 

99495 
99497 

.03 
.03 

81440 
81525 

1.40 
1.41 

43 
42 

19 

99377 

.04 

76794 

1.27 

41 

19 

99499 

.03 

81609 

1.41 

41 

20 
21 
22 

99379 
9.99381 
99383 

.04 
.04 
.04 

76870 
10.76946 
77023 

1.27 
1.27 

1.28 

40 
39 
38 

20 
21 

22 

99501 
9.99503 
99505 

.03 
.03 
.03 

81694 
10.81779 
81864 

1.41 
1.41 
1.42 

40 
39 
38 

23 
24 

99385 
99388 

.04 
.04 

77099 
77176 

1.28 
1.28 

37 
36 

23 
24 

99507 
99509 

.03 
.03 

81949 
82035 

1.42 
1.42 

37 
36 

25 
26 

27 

99390 
99392 
99394 

.04 
.04 
.04 

77253 
77330 
77407 

1.28 
1.28 
.29 

35 
34 
33 

25 

26 

27 

99511 
99513 
99515 

.03 
.03 
.03 

82120 
82206 
82292 

1.43 
1.43 
1.43 

35 
34 
33 

28 

99396 

.04 

77484 

.29 

32 

28 

99517 

.03 

82378 

1.43 

32 

29 
30 

99398 
99400 

.04 
.04 

77562 
77639 

.29 
.29 

31 

30 

29 
30 

99518 
99520 

.03 
.03 

82464 
82550 

1.44 
1.44 

31 

30 

31 

9.99402 

.04 

10.77717 

:  .29 

29 

31 

9.99522 

.03 

10.82637 

1.44 

29 

32 

99404 

.04 

n  l 

77795 

.30 

on 

28 

32 

99524 

.03 

no 

82723 

1.44 

14  C 

28 

33 

99407 

.U4 

77873 

.oU 

27 

33 

99526 

•  Uo 

82810 

.40 

27 

34 
35 
36 

99409 
99411 
9941  3 

.04 

.03 
.03 

77951 
78029 
78107 

.30 
.30 
.31 

26 
25 
24 

34 
35 

36 

99528 
99530 
99532 

.03 
.03 
.03 

82897 
82984 
83072 

1.45 
1.45 

1.46 

26 
25 
24 

37 
38 

99415 
99417 

.03 
.03 

78186 
78264 

.31 
.31 

23 
22 

37 

38 

99533 
99535 

.03 

.03 

83159 
83247 

1.46 
1.46 

23 

22 

39 

99419 

.03 

78343 

.31 

21 

39 

99537 

.03 

83335 

1.46 

21 

40 
41 

99421 
9.99423 

.03 
.03 

78422 
10.78501 

.31 

.32 

20 
19 

40 
41 

99539 
9.99541 

.03 
.03 

83423 
10.83511 

1.47 
1.47 

20 
19 

42 

99425 

.03 

78580 

.32 

18 

42 

99543 

.03 

83599 

]  .47 

18 

43 

99427 

.03 

78659 

.32 

17 

43 

99545 

.03 

83688 

1.48 

14  Q 

17 

44 

99429 

.03 

78739 

.32 

16 

44 

99546 

.03 

83776 

.48 

16 

45 

99432 

.03 

no 

78818 

.33 

OO 

15 

45 

99548 

.03 

OO 

83865 

1.48 

1  4ft 

15 

46 

47 

99434 
99436 

.Uo 

.03 

78898 
78978 

.00 

.33 

14 
13 

46 
47 

99550 
99552 

o 

.03 

83954 
84044 

1.48 

1.49 

14 
13 

48 

99438 

.03 

79058 

.33 

12 

48 

99554 

.03 

84133 

1.49 

12 

49 
50 

99440 
99442 

.03 
.03 

79138 
79218 

.34 

.34 

11 

10 

49 
50 

99556 
99557 

.03 
.03 

84223 
84312 

1.50 

1c  n 

11 
10 

51 

9.99444 

.03 

no 

10.79299 

.34 

9 

51 

9.99559 

.03 
.03 

10.84402 

.00 

1.50 

9 

52 

99446 

•  Uo 

79379 

•  o4 

8 

52 

99561 

84492 

8 

53 

99448 

.03 

no 

79460 

.34 

Q  r 

7 

53 

99563 

.03 

no 

84583 

1.51 

-i  ci 

7 

54 

99450 

.Uo 
no 

79541 

.00 

6 

54 

99565 

.Uo 
.03 

84673 

1.01 

1.51 

6 

55 

99452 

.Uo 

79622 

.00 

5 

55 

99566 

84764 

5 

56 
57 

99454 
99456 

.03 
.03 

79703 

79784 

.35 
.35 

4 
3 

56 
57 

99568 
99570 

.03 
.03 

84855 
84946 

1.51 
1.52 

4 
3 

58 

99458 

.03 

no 

79866 

.36 
i  ^fi 

2 

58 

99572 

.03 

no 

85037 

1.52 
1  i2 

2 

59 

99460 

.Uo 

79947 

i  .00 

1 

59 

99574 

.UO 
no 

85128 

J..  J-6 

1 

60 

99462 

.03 

80029 

1  .36 

0 

60 

99575 

.Uo 

85220 

1.53 

0 

~M7 

Cosine. 

Dl" 

Coiling. 

Dl" 

M. 

M. 

Cosine. 

"BT7 

Col  R  rig. 

Dl" 

M. 

9° 


SINES  AND  TANGENTS. 


88° 


M. 

Sine.   Dl" 

Tang,  i  Dl" 

BI. 

M.    Sine.   Dl" 

Tang. 

Dl" 

M. 

0 
1 

2 
3 
4 
5 
6 

9.99575 
99577 
99579 
99581 
99582 
99584 
99586 

0.03 
.03 
.03 
.03 
.03 
.03 

10.85220 
85312 
85403 
85496  i 
85588  | 
85680 
85773 

1.53 
1.53 
1.54 
1.54 
1.54 
1.55 

60 
59 

58 
57 
56 
55 
54 

0 
1 

2 
3 
4 
5 
6 

9.99675 
99677 
99678 
99680 
99681 
99683 
99684 

0.03 
.03 
.03 
.03 
.03 
.03 

10.91086 
91190 
91295 
91400 
91505 
91611 
91717 

1.74 
1.75 
1.75 
1.76 
1.76 
1.76 

60 
59 
58 
57 
56 
55 
54 

7 

99588 

.03 

85866 

1.55 

53 

7 

99686 

.03 

91823 

1.77 

53 

8 

99589 

.03 

AO 

85959 

1.55 

Ire 

52 

8 

99687 

.03 

AO 

91929 

L.77 

1  *7Q 

52 

9 
10 

99591 
99593 

.Uo 

.03 

86052 
86146 

.00 

1.56 

-I   C  I* 

51 
50 

9 
10 

99689 
99690 

.Uo 
.03 

AO 

92036 
92142 

L.7o 

1.78 

1  *7Q 

51 

50 

11 

9.99595 

.03 

10.86239  i'?" 

49 

11 

9.99692 

.U«i 

10.92249 

L.7o 

49 

12 
13 
14 

99596 
99598 
99600 

.03 
.03 
.03 

86333 
86427 
86522 

l.OO 

1.57 
1.57 

48 
47 
46 

12 
13 
14 

99693 
99695 
99696 

.03 
.03 
.03 

92357 
92464 
92572 

L.79 
1.79 
1.80 

48 
47 
46 

15 

99601 

.03 

86616  If'?' 

45 

15 

99698 

.02 

92680 

1.80 

45 

16 

99603 

.03 

86711  rjg 

44 

16 

99699 

.02 

92789 

1.81 

44 

17 
18 

99605 
99607 

.03 
.03 

868061}  •;£ 

86901  !  !'_ 

43 
42 

17 

18 

99701 
99702 

.02 
.02 

92897 
93006 

1.81 
1.81 

43 
42 

19 
20 
21 

99608 
99610 
9.99612 

.03 
.03 
.03 
no, 

86996 
87091 
10.87187 

i.oy 

1.59 
1.59 
i  «n 

41 

40 
39 

19 
20 
21 

99704 
99705 
9.99707 

.02 
.02 
.02 

93115 
93225 
10.93334 

1.82 
1.82 
1.83 

1  ft^ 

41 

40 
39 

22 

99613 

.UO 

87283  ;"'" 

38 

22 

99708 

.uz 

93444 

i»oO 

38 

23 

99615 

.03 
no 

87379  i}'™ 

37 

23 

99710 

.02 
09 

93555 

1.84 

1  84. 

37 

24 
25 

99617 
99618 

.UO 

.03 

AO 

87475  jj-jr 
87572  L1'^ 

36 
35 

24 
25 

99711 
99713 

•  UZ 

.02 

An 

93665 
93776 

1  .OTC 

1.85 

In  c 

36 
35 

26 

27 

99620 
99622 

.Uo 

.03 

87668  Hi 

87765  |H1 

34 
33 

26 
27 

99714 
99716 

.02 

.02 

93887 
93998 

3  A 

1.86  |  ™ 

28 

99624 

.03 
03 

87862 

1.02 

32 

28 

99717 

.02 

94110 

1.86 

IQfi 

32 

29 
30 
31 

99625 
99627 
9.99629 

'.03 
.03 

A-J 

87960 
88057 
10.88155 

1.63 
1.63 

1AO 

31 
30 
29 

29 
30 
31 

99718 
99720 
9.99721 

!02 
.02 

On 

94222 
94334 
10.94447 

.00 

1.87 
1.87 

Ion 

31 

30 
29 

32 

99630 

.Uo 

88253 

.Do 

28 

32 

99723 

2 

94559 

.00 

28 

33 

99632 

.03 
no 

88351 

1.64 
l  fid 

27 

33 

99724 

.02 

09 

94672 

1.88 

1QQ 

27 

34 
35 

99633 
99635 

.Uo 

.03 

88449 

88548 

1  .04 

1.64 

26 
25 

34 
35 

99726 
99727 

2 

.02 

94786 
94899 

.oy 
1.89 

26 
25 

36 

99637 

.03 
.03 

88647 

1.65 

1C  X 

24 

36 

99728 

.02 

09 

95013 

1,90 

1  Qft. 

24 

37 

99638 

88746 

•  00 

23 

37 

99730 

/ 

95127 

l.VU 

23 

38 
39 

99640 
99642 

.03 
.03 
03 

88845 
88944 

1.65 
1.66 
Ififi 

22 
21 

38 
39 

99731 
99733 

.02 
.02 
09 

^95242 
95357 

1.91 
1.91 
1  Q2 

22 
21 

40 

99643 

89044 

.DO 

20 

40 

99734 

L 

95472 

20 

41 

9.99645 

.03 

AO 

10.89144 

1.67 

IrtW 

19 

41 

9.99736 

.02 

On 

10.95587 

1.92 

Too 

19 

42 

99647 

•  Uo 

AO 

89244 

.o/ 

18 

42 

99737 

i 

95703 

.y6 

18 

43 

99648 

.Uo 

.03 

89344 

1.67 
1  fift 

17 

43 

99738 

,02 

09 

95819 

1.93 

17 

44 

99650 

89445 

l.Oo 

16 

44 

99740 

z 

95935 

1.V4 

16 

45 

99651 

.03 

89546 

1.68 

15 

45 

99741 

.02 

96052 

1.94 

15 

46 

99653 

.03 

89647 

1.68 

14 

46 

99742 

.02 

96168 

1.95 

14 

47 

99655 

.03 

v  89748 

1.69 

13 

47 

99744 

.02 

96286 

1.95 

13 

48 

99656 

.03 

A-> 

89850 

1.69 

12 

48 

99745 

.02 

96403 

1.96 

12 

49 

»9658 

.Uo 

89951 

1.70 

11 

49 

99747 

.02 

96521 

1.96 

11 

50 

99659 

.03 

90053 

1.70 

10 

50 

99748 

,02 

96639 

1.97 

10 

51 

9.99661 

.03 

10.90155 

1.70 

9 

51 

9.99749 

.0$ 

10.96758 

1.97 

9 

52 

99663 

.03 

90258 

1.71 

8 

52 

99751 

.02 

96876 

1.98 

8 

53 

99664 

.03 

90360 

1.71 

7 

53 

99752 

.02 

96995 

1.98 

7 

54 

99666 

.03 

AO 

90463 

1.71 

1-  .» 

6 

54 

99753 

.02 

On 

97115 

1.99 

6 

55 

99667 

.Uo 

90566 

.72 

5 

55 

99755 

L 

97234 

2.00 

5 

56 

99669 

.03 

90670 

1.72 

4 

56 

99756 

.02 

97355 

2.00 

4 

57 

99670 

.03 

90773 

1,73 

3 

57 

99757 

.02 

97475 

2.01 

3 

58 

99672 

.03 

90877 

1.73 

2 

58 

99759 

.02 

97596 

2.01 

2 

59 
60 

99674 
99675 

.03 
.03 

90981 
91086 

1.73 
1.74 

1 

0 

59 
60 

99760 
99761 

.02 
.02 

97717 

97838 

2.02 
2.02 

1 
0 

IT 

Cosine. 

Dl" 

Cotang. 

Tvr~ 

M. 

M. 

Cosine. 

DP 

Cotang. 

DV 

M. 

7° 


S.  N.  40. 


TABLE  IV.— LOGARITHMIC 


M. 

Sine. 

Dl" 

Tang. 

Dl" 

M. 

11. 

Sine. 

Dl" 

Tang. 

Dl" 

M. 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 

9.99761 
99763 
99764 
99765 
99767 
99768 
99769 
99771 
99772 
99773 
99775 
9.99776 
99777 
99778 
99780 
99781 
99782 

0.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 

10.97838 
97960 
98082 
98204 
98327 
98450 
98573 
98697 
98821 
98945 
99070 
10.99195 
99321 
99447 
99573 
99699 
99826 

2.03 
2.03 
2.04 
2.04 
2.05 
2.06 
2.06 
2.07 
2.07 
2.08 
2.09 
2.09 
2.10 
2.10 
2.11 
2.12 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 

0 
1 
2 

3 
4 
5 
6 

7 
8 
9 
10 
11 
12 
13 
14 
J5 
16 

9.99834 
99836 
99837 
99838 
99839 
99840 
99841 
99842 
99843 
99844 
99845 
9.99846 
99847 
99848 
99850 
99851 
99852 

0.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 

11.05805 
05951 
06097 
06244 
06391 
06538 
06687 
06835 
06984 
07134 
07284 
11.07435 
07586 
07738 
07890 
08043 
08197 

2.43 
2.44 
2.45 
2.45 

2.46 
2.47 
2.48 
2.49 
2.50 
2.50 
2.51 
2.52 
2.53 
2.54 
2.55 
2.56 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 

17 

18 
19 

99783 
99785 
99786 

.02 
.02 
.02 

99954 
11.00081 
00209 

2.12 
2.13 
2.13 

43 
42 
41 

17 

18 
19 

99853 
99854 
99855 

.02 
.02 
.02 

08350 
08505 
08660 

2.56 
2.57 

2.58 

43 
42 
41 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 

99787 
9.99788 
99790 
99791 
99792 
99793 
99795 
99796 
99797 
99798 
99800 
9.99801 
99802 
99803 
99804 
99806 
99807 
99808 
99809 
99810 
99812 
9.99813 
99814 
99815 
99816 
99817 
99819 
99820 

.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 

09 

00338 
11.00466 
00595 
00725 
00855 
00985 
01116 
01247 
01378 
01510 
01642 
11.01775 
01908 
02041 
02175 
02309 
02444 
02579 
02715 
02850 
02987 
11.03123 
03261 
03398 
03536 
03675 
03813 
03953 

2.14 
2.15 
2.15 
2.16 
2.16 
2.17 
2.18 
2.18 
2.19 
2.20 
2.20 
2.21 
2.22 
2.22 
2.23 
2.24 
2.24 
2.25 
2.26 
2.26 
2.27 
2.28 
2.29 
2.29 
2.30 
2.31 
2.31 
2.32 

200 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 

99856 
9.99857 
99858 
99859 
99860 
99861 
99862 
99863 
99864 
99865 
99866 
9.99867 
99868 
99869 
99870 
99871 
99872 
99873 
99874 
99875 
99876 
9.99877 
99878 
99879 
99879 
99880 
99881 
99882 

.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
0* 

08815 
11.08971 
09128 
09285 
09443 
09601 
09760 
09920 
10080 
10240 
10402 
11.10563 
10726 
10889 
11052 
11217 
11382 
11547 
11713 
11880 
12047 
11.12215 
12384 
12553 
12723 
12894 
13065 
13237 

2.59 
2.60 
2.61 
2.62 
2.63 
2.64 
2.65 
2.66 
2.67 
2.68 
2.69 
2.70 
2.71 
2.72 
2.73 
2.74 
2.75 
2.76 
2.77 
2.78 
2.79 
2.80 
2.81 
2.82 
2.83 
2.84 
2.85 
2.87 

2QO 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 

48 

99821 

04092 

12 

48 

99883 

09 

13409 

2  on 

12 

49 
50 

99822 
99823 

.02 

09 

04233 
04373 

2.34 

2  OR 

11 

10 

49 
50 

99884 
99885 

.02 

02 

13583 
13757 

2.90 
2Q1 

11 
10 

51 
52 

9.99824 
99825 

.02 

09 

11.04514 
04656 

2.36 

9 

8 

51 

52 

9.99886 
99887 

.02 
02 

11.13931 
14107 

2.92 

2Q-J 

9 

8 

53 

99827 

09 

04798 

9  ^7 

7 

53 

99888 

02 

14283 

2  95 

7 

54 

55 
56 
57 

99828 
99829 
99830 
99831 

.02 
.02 
.02 

04940 
05083 
05227 
05370 

2.38 
2.39 
2.40 

6 
5 
4 
3 

54 
55 
56 

57 

99889 
99890 
99891 
99891 

.02 
.01 
.01 

01 

14460 
14637 
14815 
14994 

2.96 
2.97 
2.98 

6 
5 
4 
3 

58 
59 
60 

99832 
99833 
99834 

.02 
.02 

05515 
05660 
05805 

.41 

2.41 
2.42 

2 
1 
0 

58 
59 
60 

99892 
99893 
99894 

.01 
.01 

15174 
15354 
15536 

3.01 
3.02 

2 
1 
0 

BI. 

Cosine. 

Dl" 

Cotang. 

Dl" 

31. 

M. 

Cosine.  Dl" 

Cotang. 

Dl" 

M. 

70 


86C 


SINES  AND  TANGENTS. 


87° 


M.  |  Sine. 

Dl" 

Tang. 

Dl"   M. 

M.   Sine. 

Dl" 

Tan-. 

Dl" 

M. 

0 

1 

9.99894 

99895 

0.01 

11.15536 
15718 

3.03 

60 
59 

0 
1 

9.99940 
99941 

0.01 

11.28060 
28303 

4.04 

59 

2 

99896 

.01 

15900  ™J 

58 

2 

99942 

.01 

28547 

4.06 

58 

3 

99897 

.01 
n  i 

16084  |:M£ 

57 

3 

99942 

.01 

I1  1 

28792 

4.09 
4-1  1 

57 

4 
5 
6 

99898 
'99898 
99899 

.U  1 
.01 
.01 

16268 
1  6453 
16639 

•  ).U  1 

3.08 
3.10 

56 
55 
54 

4 

5 
6 

99943 
99944 
99944 

.U  I 

.01 
.01 

29038 
29286 
29535 

.1  1 
4.13 
4.15 

56 
55 
54 

7 

99900 

.01 
01 

16825  ?'!J 

53 

7 

99945 

.01 
01 

29786 

4.18 

490 

53 

8  ' 

99901 

.  U  1 

17013  r!. 

52 

8 

99946 

.U  1 

30038 

.^U 

52 

9 

99902 

.01 

17201 

51 

9 

99946 

.01 

30292 

4.23 

51 

10 

99903 

.01 

1  7390 

3.15 

50 

10 

99947 

.01 

30547 

4.25 

50 

11 

9.99904 

.01 

11.17580 

3.16 

49 

II 

9.99948 

.01 

11.30804 

4.28 

49 

12 

99904 

.01 

A  1 

17770 

3.18 

31  O 

48 

12 

99948 

.01 

A  1 

31062 

4.30 

4QQ 

48 

13 

99905 

.(>  I 

17962 

.  1  ,1 

47 

13 

99949 

.0  1 

31322 

.GO 

47 

14 
15 

99906 
99907 

.01 
.01 

18154 

18347 

3.21 
3.22 

46 
45 

14 
15 

99949 
99950 

.01 
.01 

31583 
31846 

4.35 

4.38 

46 
45 

16 

99908 

.01 

18541 

3.23 

44 

16 

99951 

.01 

32110 

4.41 

44 

17 

18 
19 

99909 
99909 
99910 

.01 
.01 
.01 
01 

18736 
18932 
19128 

3.25 
3.26 

3.28 

o  90 

43 
42 
41 

17 
18 
19 

99951 
99952 
99952 

.01 
.01 
.01 

01 

32376 
32644 
32913 

4.43 
4.46 
4.49 

43 
42 
41 

20 
21 
22 

99911 
9.99912 
99913 

•  V  1 

.01 
.01 

0  1 

19326 
11.19524 
19723 

3^31 

3:S 

40 
39 
38 

20 
21 

22 

99953 
9.99954 
99954 

•U  1 

.01 
.01 

01 

33184 
11.33457 
33731 

4!54 
4.57 

J.  fiO 

40 
39 
38 

23 
24 
25 
26 

27 

99913 
99914 
99915 
99916 
99917 

!oi 

.01 
.01 
01 

19924 
20125 
20327 
20530 
20734 

3.35 
3.37 
3.38 
3.40 

«>  41 

37 
36 
35 
34 
33 

23 
24 
25 
26 

27 

99955 
99955 
99956 
99956 
99957 

.('  1 

.01 
.01 
.01 
.01 

(II 

34007 
34285 
34565 
34846 
35130 

4:.OU 

4.63 
4.66 
4.69 
4.72 

47^ 

37 
36 
35 
34 
33 

28 

99917 

20939 

.>  .4  I 

32 

28 

99958 

•  U  1 

35415 

.  t  0 

4*rn 

32 

29 
30 
31 
32 

99918 
99919 
9.99920 
99920 

'.01 
.01 
.01 
01 

21145 
21351 
11.21559 

21768 

3.43 
3.45 
3.46 
3.48 
^50 

31 

30 
29 

28 

29 
30 
31 

32 

99958 
99959 
9.99959 
99960 

.01 
.01 
.01 
.01 
01 

35702 
35991 
11.36282 
36574 

.78 
4.82 
4.85 

4.88 

4Q-I 

31 
30 

29 
28 

33 
34 

99921 
99922 

!oi 

21978 
22189 

3.51 

27 
26 

33 
34 

99960 
99961 

."J  1 

.01 

36869 
37166 

.  «'  1 

4.95 

27 
26 

35 
36 
37 
38 
39 
40 
41 
42 

99923 
99923 
99924 
99925 
99926 
99926 
9.99927 
99928 

.01 
.01 
.01 
.01 
.01 
.01 
.01 
.01 
01 

22400 
22613 
22827 
23042 
23258 
23475 
11.23694 
23913 

3.53 
3.55 
3.57 
3.58 
3.60 
3.62 
3.64 
3.65 

25 
24 
23 
22 
21 
20 
19 
18 

35 
36 
37 
38 
39 
40 
41 
42 

99961 
99962 
99962 
99963 
99963 
99964 
9.99964 
99965 

.01 
.01 
.01 
.01 
.01 
.01 
.01 
.01 

01 

37465 
37766 
38069 
^38374 
38681 
38991 
11.39302 
39616 

4.98 
5.02 
5.05 
5.09 
5.12 
5.16 
5.19 
5.23 

t  97 

25 
24 
23 
22 
21 
20 
19 
18 

43 
44 
45 

99929 
99929 
99930 

!oi 

.01 

A1 

24133 
24355 
24577 

3^69 
3.71 

17 
16 
15 

43 
44 
45 

99966 
99966 
99967 

•  V  I 

.01 

.01 

39932 
40251 
40572 

O.Z  i 

5.31 
5.35 

17 
16 
15 

46 

99931 

.Ul 

24801  *•'? 

14 

46 

99967 

.01 

40895 

5.39 

14 

47 

99932 

.01 

25026  IH* 

13 

47 

99967 

.01 

41221 

5.43 

13 

48 

99932 

.01 

25252 

6.1  I 

12 

48 

99968 

.01 

41549 

5.47 

12 

49 
50 

99933 
99934 

.01 
.01 

25479 
25708 

3.79 
3.81 

11 
10 

49 
50 

99968 
99969 

.01 
.01 

41879 
42212 

5.51 
5.55 

11 
10 

51 

9.99934 

.01 

A  1 

11.25937 

3.83 

30  r 

9 

51 

9.99969 

.01 

11.42548 

5.59 

5R.A 

9 

52 

99935 

.U  1 

26168 

.oD 

8 

52 

99970 

.01 

42886 

.04 

8 

53 

99936 

.01 

26400 

3.87 

7 

53 

99970 

.01 

43227 

5.68 

7 

54 

99936 

.01 

26634 

3.89 

6 

54 

99971 

.01 

43571 

5.73 

6 

55 
56 

99937 
99938 

.01 
.01 

26868 
27104 

3.91 
3.93 

5 

4 

55 
56 

99971 
99972 

.01 
.01 

43917 
44266 

5.77 
5.82 

5 
4 

57 

99938 

.01 

27341 

3.95 

3 

57 

99972 

.01 

44618 

5.87 

3 

58 

99939 

.01 

27580 

3.97 

2 

58 

99973 

.01 

44973 

5.91 

2 

59 

99940 

.01 

27819 

3.40 

1 

59 

99973 

.01 

45331 

5.96 

1 

60 

99940 

.01 

28060 

4.02 

0 

60 

99974 

.01 

45692 

6.01 

0 

M. 

Cosine. 

Dl" 

Cotang. 

Dl" 

M. 

M. 

Cosine. 

Dl" 

Cotans. 

1)1" 

M. 

71 


2 


TABLE  IV.— SIXES  AND  TANGENTS. 


89° 


M. 

Sine.   Dl"   Tang.   Dl"   M. 

M.    Sine. 

Dl" 

Tang.   1)1" 

M. 

0 

9.99974 

OA1 

11.45692 

C  A£ 

60 

0 

9.99993  i  ftn. 

11.75808 

too 

60 

1 

99974 

.01 

46055 

b.Uo 

59 

1 

99994  '"„ 

76538 

\L.L 

59 

2 
3 

99974 
99975 

.01 
.01 

46422  J'li 
46792  I'H 

58 
57 

2 
3 

99994 
99994 

.UU4 
.004 

77280  )*•* 

78036  J;-° 

58 
57 

4 

99975 

.01 

47165.  Jq: 

56 

4 

99994 

.003 

78805.  lir° 

56 

5 

99976 

.01 

47541  Sis 

55 

5 

99994 

.003 

79587 

10.  U 

55 

6 

99976 

.01 

47921  i  R'.,Q 

54 

6 

99995 

.003 

80384 

13.3 

54 

7 

99977 

.01 

A  1 

48304  J*J5 

53 

7 

99995 

.003 

81196 

13.5 

]•}  Q 

53 

8 

99977 

.u  L 

48690  i  J'jTJ 

52 

8 

99995 

.003 

82024 

•  >.O 

52 

9 

99977 

.01 

49080  !^r 

51 

9 

99995 

.003 

82867 

14.1 

51 

10 

99978 

.01 

49473  iJ'J? 

50 

10 

99995 

.003 

83727 

14.3 

50 

11 

9.99978 

.01 

11.49870  ™ 

49 

11 

9.99996 

.003 

11.84605 

14.6 

49 

12 

99979 

.01 

50271  MJ? 

48 

12 

99996 

.003 

85500 

14.9 

48 

13 
14 
15 

99979 
99979 
99980 

.01 
.01 
.01 

50675 
51083 
51495 

D./4 

6.80 
6.87 

47 
46 
45 

13 
14 
15 

99996 
99996 
99996 

.003 
.003 
.003 

86415 
87349 
88304 

15.2 
15.6 
15.9 

47 

46 
45 

16 

99980 

.01 

51911  2'™ 

44 

16 

99996 

.003 

89280 

16.3 

i  £  a 

44 

17 

99981 

.01 

52331  r22 

43 

17 

99997 

.003 

90278 

Ib.b 

43 

18 
19 
20 

99981 
99981 
99982 

.01 
.01 
.01 

52755 
53183 
53615 

42 
41 
40 

18 
19 
20 

99997 
99997 
99997 

.003 
.003 
.002 

91300  {!•" 
92347  \ltA 
93419  'I': 

42 
41 
40 

21 

22 

9.99982 
99982 

.01 
.01 

A  1 

11.54052  ••''£ 

54493  ;•?; 

39 
38 

21 
22 

9.99997 
99997 

.002 
.002 

11.94519  |j|«; 
95647  \°'° 

39 
38 

23 

99983 

.Ul 

54939  \1A* 

37 

23 

99997 

.002 

96806 

1  W.O 

37 

24 
25 
26 
27 

99983 
99983 
99984 
99984 

.01 
.01 
.01 
.01 

01 

55389  '•?" 
55844  J'J° 
56304  ^ 
56768  ,00 

36 
35 
34 
33 

24 
25 
26 
27 

99998 
99998 
99998 
99998 

.002 
.002 
.002 
.002 

ffcJIQ 

97996  |iJ-J 
W219;"'4 
12.00478  J{'" 

9im£-! 

36 
35 
34 
33 

28 
29 
30 
31 
32 
33 
34 
35 
36 

99984 
99985 
99985 
9.99985 
99986 
99986 
99986 
99987 
99987 

•  U  1 

.01 
.01 
.01 
.01 
.01 
.01 
.01 
.01 

01 

57238 
57713 
58193 
11.58679 
59170 
59666 
60168 
60677 
61191 

7.91 
8.00 
8.09 
8.18 
8.28 
8.37 
8.47 
8.57 

o  f»7 

32 
31 
30 
29 
28 
27 
26 
25 
24 

28 
29 
30 
31 
32 
33 
34 
35 
36 

99998 
99998 
99998 
9.99998 
99999 
99999 
99999 
99999 
99999 

•  UU^J 

.002 
.002 
.002 
.002 
.002 
.002 
.002 
.002 

An  i 

03111 
04490 
05914 
12.07387 
08911 
10490 
12129 
13833 
15606 

4«.O 

23.0 
23.7 
24.5 
25.4 
26.3 
27.3 
28.4 
29.5 

on  Q 

32 
31 
30 
29 
28 
27 
26 
25 
24 

37 
38 
39 
40 
41 
42 
43 
44 
45 

99987 
99988 
99988 
99988 
9.99989 
99989 
99989 
99989 
99990 

•  Ul 

.01 
.005 
.005 
.005 
.005 
.005 
.005 
.005 

61711 
62238 
62771 
63311 
11.63857 
64410 
64971 
65539 
66114 

o.O  i 

8.78 
8.88 
8.99 
9.11 
9.22 
9.34 
9.46 
9.59 

23 
22 
21 
20 
19 
18 
17 
16 
15 

37 
38 
39 
40 
41 
42 
43 
44 
45 

99999 
99999 
99999 
99999 
9.99999 
99999 
99999 
10.00000 
00000 

•  UU1 

.001 
.001 
.001 
.001 
.001 
.001 
.001 
.001 

17454 

19385 
21405 
23524 
12.25752 
28100 
30582 
33215 
36018 

Ou.o 

32.2 
33.7 
35.3 
37.1 
39.1 
41.4 
43.9 
46.7 

23 
22 
21 
20 
19 
18 
17 
16 
15 

46 

47 

99990 
99990 

.005 
.004 

66698  ig'85 

14 
13 

46 
47 

00000 
00000 

.001 
.001 

39014 
42233 

49.9 
53.6 

14 
13 

48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 

99990 
99991 
99991 
9.99991 
99992 
99992 
99992 
99992 
99992 
99993 
99993 
99993 

.004 
.004 
.004 
.004 
.004 
.004 
.004 
.004 
.004 
.004 
.004 
.004 

67888 
68495 
69112 
11.69737 
70371 
71014 
71668 
72331 
73004 
73688 
74384 
75090 

S3 

10.3 
10.4 
10.6 
10.7 
10.9 
11.1 
11.2 
11.4 
11.6 
11.8 

12 
11 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 

48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 

00000 
00000 
00000 
10.00000 
00000 
00000 
00000 
00000 
00000 
00000 
00000 
00000 

.001 
.001 
.001 
.001 
.000 
.000 
.000 
.000 
.000 
.000 
.000 

45709 
49488 
53627 
12.58203 
63318 
69118 
75812 
83730 
93421 
13.05915 
23524 
53627 

57.9 
63.0 
69.0 
76.3 
85.3 
96.7 
112 
132 
162 
208 
294 
502 

12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 

60 

99993 

.004 

75808 

12.0 

0 

60 

00000 

Infinite. 

0 

M. 

Cosine. 

Dl" 

Cot  a  n  g. 

Dl" 

31. 

M. 

Cosine. 

Dl"  1  Cotang. 

Dl" 

31. 

72 


O° 


0°-3°45' 


TRAVERSE  TABLES. 


86°15'-90C 


D, 

Lat, 

Dep, 

Lat, 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

D. 

0° 

O' 

O3 

15' 

O 

SO' 

O° 

45' 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

1.000 
2.000 
3.000 
4.000 
5.000 
6.000 
7.000 
8.000 
9.000 
10.000 

.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 
.000 

1  .000 
2.000 
3.000 
4.000 
5.000 
6.000 
7.000 
8.000 
9.000 
10.000 

.004 
.009 
.013 
.018 
022 
!026 
.031 
.035 
.039 
.044 

1  .000 
2.000 
3.000 
4.000 
5.000 
6.000 
7.000 
8.000 
9.000 
10.000 

.009 
.018 
.026 
.035 
.044 
.052 
.061 
.070 
.079 
.087 

1.000 
2.000 
3.000 
4.000 
5.000 
5.999 
6.999 
7.999 
8.999 
9.999 

.013 
.026 
.039 
.052 
.065 
.079 
.092 
.105 
.118 
.131 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

9O° 

O' 

89° 

45' 

89° 

30' 

89° 

15' 

1° 

O' 

1° 

15' 

1° 

3<y 

1° 

45' 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

1.000 
2.000 
3.000 
3.999 
4.999 
5.999 
6.999 
7.999 
8.999 
9.999 

.017 
.035 
.052 
.070 
.087 
.105. 
.122 
.140 
.157 
.174 

1.000 
2.000 
2.999 
3.999 
4.999 
5.999 
6.998 
7.998 
8.998 
9.998 

.022 
.044 
.065 
.087 
.109 
.131 
.153 
.175 
.196 
.218 

1  .000 
1.999 
2.999 
3.999 
4.998 
5.998 
6.998 
7.997 
8.997 
9.997 

.026 
.052 
.079 
.105 
.131 
.157 
.183 
.209 
.236 
.262 

1  .000 
1.999 
2.999 
3.998 
4.998 
5.997 
6.997 
7.996 
8.996 
9.995 

.031 
.061 
.092 
.122 
153 
.183 
.214 
.244 
.275 
.305 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

89  J 

O' 

88° 

45' 

88° 

3O' 

88° 

15' 

2° 

0' 

2° 

15' 

2° 

30' 

*>° 

45' 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

.999 
1.999 
2.998 
3.998 
4.997 
5.996 
6.996 
7.995 
8.995 
9.994 

.035 
.070 
.105 
.140 
.174 
209 
.244 
.279 
.314 
.349 

.999 
1.999 
2.998 
3.997 
4.996 
5.995 
6.995 
7.994 
8.993 
9.992 

.039 
.079 
.118 
.157 
.196 
.236 
.275 
.314 
.353 
.393 

.999 
1  .998 
2.997 
3.996 
4.995 
5.994 
6.993 
7.99'2 
8.991 
9.990 

.044 
.087 
.131 
.174 

.218 
.262 
.305 
.349 
.393 
.436 

.999 
1.998 
2.997 
3.995 
4.994 
5.993 
6.992 
7.991 
8.990 
9.988 

.048 
.096 
.144 
.192 

.240 
.288 
,336 
.384 
.432 
.480 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

88° 

w 

87° 

45' 

87° 

30' 

87° 

15' 

3° 

O' 

3 

15' 

3n 

3O' 

3° 

45' 

J 
2 
3 
4 
5 
6 
7 
8 
9 
10 

.999 
1.997 
2.996 
3.995 
4.993 
5.992 
6.990 
7.989 
8.988 
9.986 

.052 
.105 
.157 
.209 
.262 
.314 
.366 
.419 
.471 
.523 

.998 
1.997 
2.995 
3.994 
4.992 
5.990 
6.989 
7.987 
8.986 
9.984 

.057 
.113 
.170 
.227 
.283 
.340 
.397 
.454 
.510 
.567 

.998 
1.996 
2.994 
3.993 
4.991 
5.989 
6.987 
7.985 
8.983 
9.981 

,061 
.122 
.183 
.244 
.305 
.366 
.427 
.488 
.549 
.610 

.998 
1.996 
2.994 
3.991 
4.989 
5.987 
6.985 
7.983 
8.981 
9.979 

.065 
.131 
.196 

.262 
.327 
.392 
.458 
.523 
.589 
.654 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

87° 

O' 

86° 

45' 

86" 

30' 

86e 

15' 

D. 

Dep, 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

D. 

73 


4°-7°45' 


TRAVERSE  TABLES. 


82°15'-86° 


D, 

Lat. 

Dep, 
Or^ 

Lat, 

Dep. 

Lat, 

Dep. 

Lat. 

Dep. 

D. 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

40 

4°  15' 

4°  30' 

4°  45' 

.998 
1.995 
2.993 
3.990 
4.988 
5.986 
6.983 
7.981 
8.978 
9.976 

.070 
.140 

.209 
.279 
.349 
.418 
.488 
.558 
.628 
.698 

.997 
1.995 
2.992 
3.989 
4.986 
5.984 
6.981 
7.978 
8.975 
9.973 

.074 
.148 
.222 
.296 
.371 
.445 
.519 
.593 
.667 
.741 

.997 
1.994 
2.991 
3.988 
4.985 
5.981 
6.978 
7.975 
8.972 
9.969 

.078 
.157 
.235 
.314 
.392 
.471 
.549 
.628 
.706 
.785 

.997 
1.993 
2.990 
3.986 
4.983 
5.979 
6.976 
7.973 
8.969 
9.966 

.083 
.166 
.248 
.331 
.414 
.497 
.580 
.662 
.745 
'  .828 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

S«°  O' 

85°  45' 

85°  3O' 

85°  15' 

5    0' 

5°  15' 

5°  3O' 

5°  45' 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

.996 
1.992 
2.989 
3.985 
4.981 
5.977 
6.973 
7.970 
8.966 
9.962 

.087 
.174 
.261 
,349 
.436 
,523 
.610 
.697 
.784 
.872. 

.996 
1.992 
2.987 
3.983 
4.979 
5.975 
6.971 
7.966 
8.962 
9.958 

.092 
.183 
.275 
.366 
.458 
.549 
.641 
.732 
.824 
.915 

.995 
1.991 
2.986 
3.982 
4.977 
5.972 
6.968 
7.963 
8.959 
9.954 

.096 
.192 
.288 
.383 
.479 
,575 
.671 
.767 
.863 
.958 

.995 
1.990 
2.985 
3.980 
4.975 
5.970 
6.965 
7.960 
8.955 
9.950 

.100 
.200 
.301 
.401 
,501 
.601 
.701 
.802 
.902 
1.002 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

85°  O' 

84°  45' 

84°  SO' 

84C  15' 

•~F 

2 
3 
4 
5 

6 

7 
8 
9 
10 

~T 

2 
8 

4 
5 
6 

7 
8 
9 
10 

6°  O' 

6°  15' 

6°  30' 

6°  45' 

T 

2 
3 
4 
5 
6 
7 
8 
9 
10 

.995 
1.989 
2.984 
3.978 
4.973 
5.967 
6.962 
7.956 
8.951 
9.945 

.105 
.209 
.314 
.418 
.523 
.627 
.732 
.836 
.941 
1.045 

.994 

1.988 
2.982 
3.976 
4.970 
5.964 
6.958 
7.952 
8.947 
9.941 

.109 

.218 
.327 
.435 
.544 
.653 
.762 
.871 
.980 
1.089 

.994 
1.987 
2.981 
3.974 
4.968 
5.961 
6.955 
7.949 
8.942 
9.936 

.113 
.226 
.340 
.453 
,566 
.679 
.792 
.906 
1.019 
1.132 

.993 
1.986 
2.979 
3.972 
4.965 
5.958 
6.952 
7.945 
8.938 
9.931 

.118 
.235 
.353 
.470 
,588 
.705 
.823 
.940 
1.058 
1.175 

84°  O' 

83°  45' 

83°  30' 

83    15' 

7°  0' 

7°  15' 

7°  SO' 

7D  45' 

.993 
1.985 
2.978 
3.970 
4.963 
5.955 
6.948 
7.940 
8.933 
9.925 

.122 
.244 

,366 
.487 
.609 
.731 
.853 
.975 
1.097 
1.219 

.992 
1.984 
2.976 
3.968 
4.960 
5.952 
6.944 
7.936 
8.928 
9.920 

.126 
.252 
.379 
.505 
.631 
.757 
.883 
1.010 
1.136 
1.262 

.991 
1.983 
2.974 
3.966 
4.957 
5.949 
6.940 
7.932 
$.923 
9.914 

.131 
.261 

,392 
,522 
.653 
.783 
.914 
1.044 
1.175 
1  .305 

.991 
1  .982 
2.973 
3.963 
4.954 
5.945 
6.936 
7.927 
8.918 
9.909 

.135 
.270 
.405 
,539 
.674 
.809 
.944 
1.079 
1.214 
1.349 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

IT 

IT 

83°  0' 

82°  45' 

82°  30' 

82  =  15' 

Dep. 

Lat, 

Dep. 

Lat. 

Dep, 

Lat. 

Dep. 

Lat. 

74 


TRAVERSE  TABLES. 


78°15'-82° 


D. 

Lat, 

Dep, 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

D, 

8° 

<y 

8° 

15' 

8° 

30' 

8 

45' 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

.990 
1.981 
2.971 
3.961 
4.951 
5.942 
6.932 
7.922 
8.912 
9.903 

.139 
.278 
.418 
.557 
.696 
.835 
.974 
1.113 
1  .253 
1.392 

.990 
1.979 
2.969 
3.959 
4.948 
5.938 
6.928 
7.917 
8.907 
9.897 

.143 

.287 
.431 
.574 
.717 
.861 
1.004 
1.148 
1.291 
1.435 

.989 
1.978 
2.967 
3.956 
4.945 
5.934 
6.923 
7.912 
8.901 
9.890 

.148 

.296 
.443 
,591 
.739 
.887 
1  .035 
1.182 
1,330 
1.478 

.988 
1.977 
2.965 
3.953 
4.942 
5.930 
6.919 
7.907 
8.895 
9.884 

.152 
.304 
.456 
.608 
.761 
.913 
1  .065 
.217 
1  .369 
1,521 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

82"' 

O' 

81° 

45' 

81" 

SO' 

81 

15 

9 

o 

9 

15' 

9 

30' 

9 

45' 

1 
2 
3 
4 

5 
6 

7 
8 
9 
10 

.988 
1.975 

2.963 
3.951 

4.938 
5.926 
6.914 

7.902 
8.889 
9.877 

.156 
.313 
.469 
.626 
.782 
.939 
1.095 
1.251 
1.408 
1  .564 

.987 
1.974 

2.961 
3.948 
4.935 
5.922 
6.909 
7.896 
8.883 
9.870 

.161 
.321 

.482 
.643 
.804 
.964 
1.125 
.286 
.447 
1.607 

.986 
1.973 
2.959 
3.945 
4.931 
5.918 
6.904 
7.890 
8.877 
9.863 

.165 
.330 
.495 
.660 
.825 
.990 
1.155 
1.320 
1  .485 
1.650 

.986 
1.971 
2.957 
3.942 
4.92S 
5.914 
6.899 
7.884 
8.870 
9.856 

.169 
.339 
,508 
.677 
.847 
.016 
.185 
.355 
,524 
.693 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

81? 

O' 

M> 

45 

so0 

30 

8O 

15' 

NT 

O' 

1O 

15 

10° 

3O 

10- 

45' 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

.985 
1.970 
2.954 
3.939 
4.924 
5.909 
6.894 
7.878 
8.863 
9.848 

.174 
.347 
.521 

.695 
.868 
1.042 
1.216 
1.389 
1,563 
1.736 

.9*4 
1.968 
2.952 
3.936 
4.920 
5.904 
6.888 
7.872 
8.856 
9.840 

.178 

.356 
.534 
.712 
.890 
1.068 
1.246 
1.424 
1.601 
1.779 

.983 
1.967 
2.950 
3.933 
4.916 
5.900 
6.883 
7.866 
8.849 
9.833 

.182 
,364 
,547 
729 
.911 
1.093 
1.276 
1.458 
1.640 
1.822 

.982 
1  .965 
2.947 
3.930 
4.912 
5.89.5 
6.877 
7.860 
8.842 
9.825 

.187 
.373 
,560 
.746 
.933 
.119 
.306 
.492 
.679 
.865 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

80 

o 

79 

45' 

79^ 

30' 

79" 

15 

11" 

0' 

11 

15' 

11° 

30' 

11° 

45' 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

.982 
1,963 
2.945 
3.927 
4.908 
5.890 
6.871 
7.853 
8.835 
9.816 

.191 
.382 
.572 
.763 
.954 
1.145 
1.336 
1,526 
1.717 
1.908 

.981 
1.962 
2.942 
3.923 
4.904 
5.885 
6.866 
7.846 
8.827 
9.808 

.195 
.390 

.585 
.780 
.976 
.171 
.366 
.561 
.756 
.951 

.980 
1.960 
2.940 
3.920 

4.900 
5.880 
6.860 
7.839 
8.819 
9.799 

.199 
.399 
.59* 
.797 
.997 
1.196 
1.396 
1,595 
1.794 
1  .994 

.979 
1.958 
2.937 
3.916 
4.895 
5.874 
6.853 
7.832 
8.811 
9.790 

.204 
.407 
.611 
.815 
.018 
.222 
.426 
.629 
.833 
2.036 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

79 

O' 

78° 

45' 

78° 

30' 

78° 

15' 

D. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

DeP. 

Lat. 

D. 

75 


12°-15°45' 


TRAVERSE  TABLES. 


74°15'-78< 


D, 

Lat. 

Dep. 

Lat, 

Dep. 

Lat, 

Dei). 

Lat. 

Dep, 

D. 

12 

O' 

12° 

15' 

12° 

3W 

12° 

45' 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

.978 
1.956 
2.934 
3.913 

4.891 
5.869 
6.847 

7.825 
8.803 
9.781 

.208 
.416 
.624 
.832 
1.040 
1.247 
1.455 
1.663 
1.871 
2.079 

.977 
1.954 
2.932 
3.909 

4.886 
5.863 
6.841 
7.818 
8.795 
9.772 

.212 
.424 
.637 
.849 
1.061 
1.273 
1.485 
1.697 
1.910 
2.122 

.976 
1.953 

2.929 
3.906 

4.882 
5.858 
6.834 
7.810 
8.787 
9.763 

.216 
.433 
.649 
.866 
1.082 
1.299 
,515 
.731 
.948 
2.164 

.975 
1.951 

2.926 
3.901 

4.877 
5.852 
6.827 
7.803 
8.778 
9.753 

.221 
.441 

.662 
.883 
.103 
.324 
,545 
1.766 
.986 
2.207 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

78e 

o 

77° 

45' 

77° 

3O' 

77° 

15' 

13 

o 

13° 

15' 

13° 

3O' 

13° 

45' 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

.974 
1.949 
2.923 

3.897 
4.872 
5.846 
6.821 
7.795 
8.769 
9.744 

.225 
.450 
.675 
.900 
1.125 
1.350 
1.575 
1.800 
2.025 
2.250 

.973 
1.947 
2920 
3.894 
4.867 
5.840 
6.814 
7.787 
8.760 
9.734 

.229 
.458 
.688 
.917 
1.146 
1.375 
1.604 
1.834 
2.063 
2.292 

.972 
1.945 
2.917 

3.889 
4.862 
5.834 
6.807 
7.779 
8.751 
9.724 

.233 
.467 
.700 
.934 
1.167 
1.401 
1.634 
1.868 
2.101 
2.334 

.971 
1.943 
2.914 

3.885 
4.857 
5.828 
6.799 
7.771 
8.742 
9.713 

.238 
.475 
.713 
.951 
1.188 
1.426 
1.664 
1.901 
2.139 
2.377 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

77° 

O' 

76° 

45' 

70° 

3O' 

7tt- 

15' 

14° 

0' 

14° 

15' 

14° 

3O' 

14° 

45' 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

.970 
1.941 
2.911 

3.881 
4.851 
5.822 
6.792 
7.762 
8.733 
9.703 

.242 
.484 
.726 
.968 
1.210 
1.452 
1.693 
1  .935 
2.177 
2.419 

.969 
r.938 
2.908 
3.877 
4.846 
5.815 
6.785 
7.754 
8.723 
9.692 

.246 
492 
.738 
.985 
1.231 
1.477 
1.723 
1.969 
2.215 
2.462 

.968 
1.936 
2.904 
3.873 
4.841 
5.809 
6.777 
7.745 
8.713 
9.681 

.250 
.501 
.751 
1.002 
1.252 
1.502 
1.753 
2.003 
2.253 
2,504 

.967 
1.934 

2.901 
3.868 
4.835 
5.802 
6.769 
7.736 
8.703 
9.670 

.255 

.509 
.764 
1.018 
1.273 

1  ,528 
1.782 
2.037 
2.291 
2.546 

1 

2 
3 
4 
5 

6 

7 
8 
9 
10 

70° 

O' 

75° 

45' 

75° 

30' 

75° 

15' 

15° 

O' 

15° 

15' 

15° 

3O' 

15° 

45' 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

.966 
1.932 
2.898 
3.864 
4.830 
5.796 
6.761 
7.727 
8.693 
9.659 

.259 
.518 
.776 
1.035 
1  .294 
1,553 
1.812 
2.071 
2.329 
2,588 

.965 
1.930 
2.894 
3.859 
4.824 
5.789 
6.754 
7.718 
8.683 
9.648 

.263 
,526 
.789 
1.052 
1.315 
1,578 
1.841 
2.104 
2.367 
2.631) 

.964 
1.927 

2.891 
3.855 
4.818 
5.782 
6.745 
7.709 
8.673 
9.636 

.267 
,534 
.802 
1.069 
1.336 
1.603 
1.871 
2.138 
2.405 
2.672 

.962 
1.925 
2887 
3.850 
4.812 
5.775 
6.737 
7.700 
8.662 
9.625 

.271 
,543 
.814 
1.086 
1,357 
1.629 
1  .900 
2.172 
2.443 
2.714 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

75° 

<y 

74° 

45' 

74° 

3O' 

74" 

15' 

D, 

Dep. 

Lat, 

Dep. 

Lat. 

Dep, 

Lat. 

Dep. 

Lat. 

D. 

16°-19°45' 


TRAVERSE  TABLES. 


70*15-74* 


D, 

"I 

2 
3 
4 
5 
6 
7 
8 
9 
10 

~T 

2 
3 
4 
5 
6 
7 
8 
9 
10 

Lat, 

Dei>. 

Lat, 

Dep. 

Lat, 

Dep, 

Lat, 

Dep. 

D. 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

16    O' 

16°  15' 

16°  30' 

16°  45' 

.961 
1.923 
2.884 
3.845 
4.806 
5.768 
6.729 
7.690 
8.651 
9.613 

.276 
.551 
.827 
1.103 
1.378 
1.654 
1.929 
2.205 
2.481 
2.756 

.960 
1.920 
2.880 
3.840 
4.800 
5.760 
6.720 
7.680 
8.640 
9.600 

.280 
.560 
.839 
1.119 
1.399 
1.679 
1.959 
2.239 
2.518 
2.798 

.959 
1.918 
2.876 
3-.835 
r4.794 
5.753 
6.712 
7.671 
8.629 
9.588 

.284 
.568 
.852 
1.136 
1.420 
1.704 
1.988 
2.272 
2.556 
2.840 

.958 
1.915 
2.873 
3.830 
4.788 
5.745 
6.703 
7.661 
8.618 
9.576 

.288 
.576 
.865 
1.153 
1.441 
1.729 
2.017 
2.306 
2.594 
2.882 
is5 

74°  O' 

73°  45' 

73°  3O' 

733 

17~  O' 

17°  15' 

17°  30' 

17°  45' 

T 

2 
3 
4 
5 
6 
7 
8 
9 
10 

.956 
1.913 
2.869 
3.825 
4.782 
5.738 
6.694 
7.650 
8.607 
9.563 

.292 
.585 
.877 
1.169 
1.462 
1.754 
2.047 
2.339 
2.631 
2.924 

.955 
1.910 
2.865 
3.820 
4.775 
5.730 
6.685 
7.640 
8,595 
9.550 

.297 
.593 
.890 
1.186 
1.483 
1.779 
2.076 
2.372 
2.669 
2.965 

.954 
1.907 
2.861 
3.815 
4.769 
5.722 
6.676 
7.630 
8.583 
9.537 

.301 
.601 
.902 
1.203 
1.504 
1.804 
2.105 
2.406 
2.707 
3.007 

.952 
1.905 
2.857 
3.810 
4.762 
5.714 
6.667 
7.619 
8.572 
9.524 

.305 
.610 
.915 
1.219 
1.524 
1.829 
2.134 
2.439 
2.744 
3.049 

73    0' 

73-'  45' 

72-'  3O' 

72°  15' 

~T 

2 
3 
4 
5 
6 
7 
8 
9 
10 

~T 

2 
3 
4 
5 
6 
7 
8 
9 
10 

D: 

18°  O' 

18°  15' 

18°  30' 

18°  45' 

.951 
1.902 
2.853 
3.804 
4.755 
5.706 
6.657 
7.608 
8,559 
9^511 

.30J 
.618 
.927 
1.236 
1.545 
1.854 
2.163 
2.472 
2.781 
3.090 

.950 
1.899 
2.849 
3.799 
4.748 
5.698 
6.648 
7.598 
8.547 
9.497 

.313 
.626 
.939 
1.253 
1.566 
1.879 
2.192 
2.505 
2.818 
3.132 

.948 
1.897 
2.845 
3.793 
4.742 
5.690 
6.638 
7.587 
8.535 
9.483 

.317 
.635 
.952 
1.269 

1.587 
1.904 
2.221 
2.538 
2.856 
3.173 

.947 

1.894 
2.841 
3.788 
4.735 
5-.6S2 
6.628 
7.575 
8.522 
9.469 

.321 
.643 
.964 
1.286 
1.607 
1.929 
2.250 
2.572 
2.893 
3.214 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

72°  O' 

71°  45' 

71°  30' 

71    15' 

19    O' 

19°  15' 

19°  30* 

19°  45' 

T 

2 
3 
4 
5 
6 
7 
8 
9 
10 

.946 
1.891 
2.837 
3.782 
4.728 
5.673 
6.619 
7,564 
8.510 
9.455 

.326 
.651 
.977 
1.302 
1.628 
1.953 
2.279 
2.605 
2.930 
3.256 

.944 

1.888 
2.832 
3.776 
4.720 
5.665 
6.609 
7.553 
8.497 
9.441 

.330 
.659 
.989 
1.319 
1.648 
1.978 
2.308 
2.638 
2.967 
3.297 

.943 
1  .885 
2.828 
3.771 
4.713 
5.656 
6.598 
7.541 
8.484 
9.426 

.334 
.668 
1.001 
1.335 
1.669 
2.003 
2.337 
2.670 
3.004 
3.338 

.941 

1.882 
2.824 
3.765 
4.706 
5.647 
6.588 
7.529 
8.471 
9.412 

.338 
.676 
1.014 
1.352 
1.690 
2.027 
2.365 
2.703 
3.041 
3.379 

71°  0' 

7O°  45' 

70°  30' 

70°  15' 

Dep. 

Lat, 

Dep. 

Lat. 

Dep, 

Lat. 

Dep. 

Lat. 

D, 

S.N.41. 


77 


20°-23°45' 


TRAVERSE  TABLES. 


°15'-70° 


D. 

Lat, 

Dep. 

o1 

Lat. 

Dep. 

Lat, 

Dep. 

Lat. 

Dep. 

D. 

~T 

2 
3 
4 
5 
6 
7 
8 
9 
10 

i 

2 
3 
4 
5 
6 
7 
8 
9 
10 

1 

2 
3 
4 

5 
6 

7 
8 
9 
10 

20° 

20-  15' 

2O    3O' 

2O     45' 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

.940 
1.879 
2.819 
3.759 
4.698 
5.638 
6.578 
7,518 
8.457 
9.397 

.342 
.684 
1.026 
1.368 
1.710 
2.062 
2.394 
2.736 
3.078 
3.420 

.938 
1.876 
2.815 
3.753 
4.691 
5.629 
6.567 
7.506 
8.444 
9.382 

.346 
.692 
1.038 
1/384 
1.731 
2.077 
2.423 
2.769 
3.115 
3.461 

.937 
1.873 
2.810 
3.747 
4.683 
5.620 
6.557 
•7.493 
8.430 
9.367 

,350 
.700 
1.051 
1.401 
1.751 
2.101 
2.451 
2.802 
3.152 
3.502 

.935 

1.870 
2.805 
3.740 
4.676 
5.611 
6.546 
7.481 
8.416 
9,351 

.354 
.709 
1.063 
1.417 
1.771 
2.126 
2.480 
2.834 
3.189 
3.543 

70°  0 

69    45 

69°  30' 

69     15' 

21     0 

213  15' 

21°  30' 

21     45' 

"I 
2 
3 
4 
5 
6 
7 
8 
9 
10 

.934 
1.867 
2.801 
3.734 
4.668 
5.601 
6,535 
7.469 
8.402 
9,336 

.358 
.717 
1.075 
1.433 
1.792 
2.15!) 
2.509 
2.867 
3.225 
3.584 

,932 

1.864 
2.796 
3.728 
4.660 
5.592 
6.524 
7.456 
8.388 
9.320 

.362 
.725 
1.087 
1.450 
1.812 
2.175 
2.537 
2.900 
3.262 
3.624 

.930 
1.861 
2.791 
3.722 
4.652 
5.582 
6,513 
7.443 
8.374 
9,304 

,367 
.733 
1.100 
1.466 
1.833 
2.199 
2.566 
2.932 
3.299 
3.665 

.929 
1.858 
2.786 
3.715 
4.644 
5.573 
6.502 
7.430 
8.359 
9.288 

.371 
.741 
1.112 
1.482 
1.853 
2.223 
2.594 
2.964 
3.335 
3.706 

«1F  O' 

68    45' 

68    3O' 

68°  15' 

22°  O' 

22°  15' 

22°  SO' 

22°  45' 

T 

2 
3 
4 
5 
6 
7 
8 
9 
10 

.927 
1.854 
2.782 
3.709 
4.636 
5.563 
6.490 
7.418 
8.345 
9.272 

.375 
.749 
1.124 
1.498 
1.873 
2.248 
2.622 
2.997 
3.371 
3.746 

.926 
1.851 
2.777 
3.702 
4.628 
5.553 
6.479 
7.404 
8.330 
9.255 

.379 
-:>- 
1.136 
1.515 
1.893 
2.272 
2.651 
3.029 
3.408 
3.786 

.924 
1.848 
2.772 
3.696 
4.619 
5.543 
6.467 
7,391 
8.315 
9.239 

.383 
.765 
1.148 
1.531 
1.913 
2.296 
2.679 
3.062 
3.444 
3.827 

.y±j 
1.844 
2.767 
3.689 
4.611 
5.533 
6.455 
7.378 
8.300 
9.222 

.387 

.773 
1.160 
1.547 
1.934 
2.320 
2.707 
3.094 
3.480 
3  867 

68°  O' 

67°  45' 

67°  30' 

67C  15' 

23°  O' 

23     15 

23°  3O' 

23=  45' 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

IT 

.921 
1.841 
2.762 
3.682 
4.603 
5.523 
6.444 
7.364 
8.285 
9.205 

.391 
.781 
1.172 
1,563 
1.954 
2.344 
2.735 
3.126 
3.517 
3.907 

.919 
1.838 
2.756 
3.675 
4.594 
5.513 
6.432 
7.350 
8.269 
VI.  1  SS 

.395 
.789 
1.184 
1.579 
1.974 
2.368 
2:763 
3.158 
3.553 
3.947 

.917 
1.834 
2.751 
3.668 
4,585 
5.502 
6.419 
7.336 
8.254 
9.17L 

.399 
.797 
1.196 
1.595- 
1.994 
2.392 
2.791 
3.190 
3.589 
3.987 

.915 
1.831 
2.746 
3.661 
4.577 
5.492 
6.407 
7.322 
8.238 
9.153 

.403 
.805 
1.208 
1.611 
2.014 
2.416 
2.819 
3.222 
3.625 
4.027 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

"DT 

6  7 
Dep. 

O' 

Lat. 

66 

Dep. 

45' 

66^  3O' 

66  '  15' 

Lat. 

Dep, 

Lat. 

Dep.    |    Lat. 

78 


24°-27°45' 


TRAVERSE  TABLES. 


62°15'-66 


D, 

Lat, 

Dep, 

Lat, 

Dep. 

Lat, 

Dep. 

Lat, 

Dep, 

D. 

~T 
2 

3 
4 
5 
6 

7 
8 
9 
10 

T 

2 
3 
4 
5 

6 
7 
8 
9 
10 

~T 

2 
3 
4 
5 
6 
7 
8 
9 
10 

24    <y 

24    15' 

24    3O' 

24    45' 

.914 
1.827 
2.741 
3.654 
4.5G8 
5.481 
6.395 
7.308 
8.222 
9.135 

.407 
.813 
1.220 
1.627 
2.034 
2.440 
2.847 
3.254 
3.66L 
4.067 

.'Jl2 
1.824 
2.735 
3.647 
4.559 
5.471 
6.382 
7.294 
8.206 
9.118 

.411 
.821 
1.232 
1.643 
2.054 
2.464 
2.875 
3.286 
3.696 
4.107 

.910 
1.820 
2.730 
3.640 
4.550 
5.460 
6.370 
7.280 
8.190 
9.100 

.415 

.829 
1.244 
1.659 
2.073 
2.488 
2.903 
3.318 
3.732 
4.147 

.908 
1.816 
2.724 
5.633 
4.541 
5.449 
6.357 
7.265 
8.173 
9.081 

.419 

.837 
1.256 
1.675 
2.093 
2.512 
2.931 
3.349 
3.768 
4.187 

«6°  w 

05    45' 

65°  30' 

65    15' 

~T 

2 
3 

4 
5 
G 
7 
8 
9 
10 

"T 

2 
3 
4 
5 

6 

7 
8 
9 
10 

25°  O' 

25"  15' 

25  so' 

25°  45' 

.9U6 
1.813 
2.719 
3.625 
4.532 
5.438 
6.344 
7.250 
8.157 
9.063 

.423 

.845 
1.268 
1.690 
2.113 
2.536 
2.958 
3.381 
3.804 
4.226 

.904 
1.809 
2.713 
3.618 
4.522 
5.427 
6.331 
7.236 
8.140 
9.045 

.427 

.853 
1.280 
1.706 
2.133 
2,559 
2.986 
3.413 
3.839 
4.266 

.903 
1.805 
2.708- 
3.610 
4.513 
5.416 
6.318 
7.221 
8.123 
9.026 

.431 
.861 
1.292 
1.722 
2.153 
2,583 
3.014 
3.444 
3.875 
4.305 

.901 
1.801 
2.702 
3.603 
4,504 
5.404 
6,305 
7.206 
8.106 
9.007 

.434 

.869 
1.303 
1.738 
2.172 
2.607 
3.041 
3.476 
3.910 
4.344 

65a  O' 

64    45' 

64    3<y 

64°  15' 

26    O' 

26°  15' 

26    SO7 

26°  45' 

"T 

2 
3 

4 
5 
6 

7 
8 
9 
10 

T 

2 
3 
4 
5 
6 
7 
8 
9 
10 

.899 
1.798 
2.696 
3.595 
4.494 
5.393 
6.292 
7.190 
8.089 
8.988 

.438 
.877 
1.315 
1.753 
2.192 
2.630 
3.069 
3.507 
3.945 
4.384 

.897 
1.794 
2.691 
3.587 
4.484 
5.381 
6.278 
7.175 
8.072 
8.969 

.442 
.885 
1.327 
1.769 
2.211 
2.654 
3.096 
3,538 
3.981 
4.423 

.895 
1.790 
2.685 
3.580 
4.475 
5.370 
6.265 
7.159 
8.054 
8.949 

.446 
.892 
1.339 
1.785 
2.231 
2.677 
3.123 
3,570 
4.016 
4.462 

.893 
1.786 
2.679 
3.572 
4.465 
5.358 
6.251 
7.144 
8.037 
8.930 

.450 
.900 
1.350 
1.800 
2.250 
2.701 
3.151 
3.601 
4.051 
4.501 

64    0' 

OS'1  45' 

63    3O' 

63    15' 

~T 

2 
3 
4 
5 
6 
7 
8 
9 
10 

DT 

27    0' 

27  r  15' 

27    SO' 

27°  45' 

.891 
1.782 
2.673 
3.564 
4.455 
5.346 
6.237 
7.128 
8.019 
8.910 

.454 
.908 
1.362 
1.816 
2.270 
2.724 
3.178 
3.632 
4.086 
4,540 

.889 
1.778 
2.667 
3,556 
4.445 
5.334 
6.223 
7.112 
8.001 
8.890 

.458 
.916 
1.374 
1.831 
2.289 
2.747 
3.205 
3.663 
4.121 
4,579 

.887 
1.774 
2.661 
3,548 
4.435 
5,322 
6.209 
7.096 
7.983 
8.870 

.462 
.923 
1.385 
1.847 
2.309 
2.770 
3.232 
3.694 
4.156 
4.617 

.885 
1.770 
2.655 
3.540 
4.425 
5.310 
6.195 
7.080 
7.965 
8.850 

.466 
.931 
1.397 
1.862 
2.328 
2.794 
3.259 
3.725 
4.190 
4.656 

63°  O' 

62°  45' 

62°  30' 

62°  15' 

Dep, 

Lat. 

Dep, 

Lat. 

Dep. 

Lat. 

Dep, 

Lat, 

D, 

79 


28°-31°45' 


TRAVERSE  TABLES. 


58°15'-62( 


D, 

T 

c 

4 

tJ 

4 

P 

e 

7 
8 
9 
10 

Lat, 

Dep, 

Lat, 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

D, 

~f 
2 
3 
4 
5 
6 
7 
8 
9 
10 

~T 

2 
3 
4 
5 
6 
7 
8 
9 
10 

T 

2 
3 
4 
5 

6 

7 
8 
9 
10 

~T 

2 
3 

4 
5 
6 

7 
8 
9 
10 

28    O* 

28    15' 

28    30' 

28°  45' 

.883 
1.766 
2.649 
3.532 
4.415 
5.298 
6.181 
7.064 
7.947 
8.829 

.469 
.939 
•1.408 
1.878 
2.347 
2.817 
3.286 
3.756 
4.225 
4.695 

.881 
1.762 
2.643 
3.524 
4.404 
5.285 
6.166 
7.047 
7.928 
8.809 

.473 
.947 
1.420 

1.S93 
2.367 
2.840 
3.313 
3.787 
4.260 
4.733 

.879 
1.758 
2.636 
3.515 
4.394 
5.273 
6.152 
7.031 
7.909 
8.788 

.477 
.954 
1.431 
1.909 
2.386 
2.863 
3.340 
3.817 
4.294 
4.772 

.877 
1.753 
2.630 
3.507 
4.384 
5.260 
6.137 
7.014 
7.890 
8.767 

.481 
.962 
1.443 
1.924 
2.405 
2.886 
3.367 
3.848 
4.329 
4.810 

62    0' 

61°  45' 

61°  3O' 

61-  15' 

29°  0' 

29    15' 

29°  3O' 

29°  45' 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

.875 
1.749 
2.624 
3.498 
4.373 
5.248 
6.122 
6.997 
7.872 
8.746 

.485 
.970 
1.454 
1.939 
2.424 
2.909 
3.394 
3.878 
4.363 
4.848 

.872 
1.745 
2.617 
3.490 
4.362 
5.235 
6.107 
6.980 
7.852 
8.725 

.489 
.977 
1.466 
1.954 
2.443 
2.932 
3.420 
3.909 
4.398 
4.886 

.870 
1.741 
2.611 
3.481 
4.352 
5.222 
6.092 
6.963 
7.833 
8.704 

.492 
.985 
1.477 
1.970 
2.462 
2.954 
3.447 
3.939 
4.432 
4.924 

.868 
1.736 
2.605 
3.473 
4.341 
5.209 
6.077 
6.946 
7.814 
8.682 

.496 
.992 
1.489 
1.985 
2.481 
2.977 
3.473 
3.970 
4.466 
4.962 

61°  O' 

«>o    45' 

6O    3O' 

6O    15' 

~T 

2 
3 
4 
5 
6 
7 
8 
9 
10 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

30°  0' 

30°  15' 

3O    3O> 

30°  45' 

.866 
1.732 
2.598 
3.464 
4.330 
5.196 
6.062 
6.928 
7.794 
8.660 

.500 
1.000 
1.500 
2.000 
2.500 
3.000 
3.500 
4.000 
4.500 
5.000 

.864 
1.728 
2.592 
3.455 
4.319 
5.183 
6.047 
6.911 
7.775 
8.638 

.504 
1.008 
1.511 
2.015 
2.519 
3.023 
3.526 
4.030 
4.534 
5.038 

.862 
1.723 
2.585 
3.446 
4.308 
5.170 
6.031 
»  6.893 
7.755 
8.616 

.508 
1.015 
1.523 
2.030 
2,538 
3.045 
3,553 
4.060 
4.568 
5.075 

.859 
1.719 
2.578 
3.438 
4.297 
5.156 
6.016 
6.875 
7.735 
8,594 

.511 
1.023 
1.534 
2.045 
2.556 
3.068 
3.579 
4.090 
4.602 
5.113 

6O°  O' 

59-  45' 

59    3O' 

59°  15' 

31°  W 

31°  15' 

31°  30' 

31°  45' 

.857 
1.714 
2.572 
3.429 
4.286 
5.143 
6.000 
6.857 
7.715 
8,572 

.515 
1.030 
1.545 
2.060 
2.575 
3.090 
3.605 
4.120 
4.635 
5.150 

.855 
1.710 
2.565 
3.420 
4.275 
5.129 
5.984 
6.839 
7.694 
8.549 

.519 
1.038 
1.556 
2.075 
2.594 
3.113 
3.631 
4.150 
4.669 
5.188 

.853 
1.705 
2.558 
3.411 
4.263 
5.116 
5.968 
6.821 
7.674 
8,526 

,522 
1.045 
1,567 
2.090 
2.612 
3.135 
3.657 
4.180 
4.702 
5.225 

.850 
1.701 
2,551 
3.401 
4.252 
5.102 
5.952 
6.803 
7.653 
8,504 

.526 
1.052 
1.579 
2.105 
2.631 
3.157 
3.683 
4.210 
4.736 
5.262 

59°  O' 

58°  45' 

58°  30* 

58°  15' 

"oT 

D. 

Dep. 

Lat, 

Dep, 

Lat. 

Dep, 

Lat. 

Dep, 

Lat. 

80 


32°-35°45' 


TRAVERSE  TABLES. 


54°15/-58° 


D, 

Lat, 

Dep. 

Lat, 

Dep. 

Lat, 

Dep, 

Lat. 

Dep, 

D, 

32°  O' 

32°  15' 

32°  3O' 

32     45' 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

~7 

2 
3 
4 
5 
6 
7 
8 
9 
10 

.848 
1.696 
2.544 
3.392 
4.240 
5.088 
5.936 
6.784 
7.632 
8.480 

.530 
1.060 
1.590 
2.120 
2.650 
3.180 
3.709 
4.239 
4.769 
5.299 

.846 
1.691 
2.537 

3.383 
4.229 
5.074 
5.920 
6.766 
7.612 
8.457 

.534 
1.067 
1.601 
2.134 
2.668 
3.202 
3.735 
4.269 
4.802 
5.336 

.843 
1.687 
2.530 
3.374 
4.217 
5.060 
5.904 
6.747 
7.591 
8.434 

.537 
1.075 
1.612 
2.149 
2.686 
3.224 
3.761 
4.298 
4.836 
5.373 

.841 
1.682 
2.523 
3.364 
4.205 
5.046 
5.887 
6.728 
7.569 
8.410 

.541 
1.082 
1.623 
2.164 

2.705 
3.246 
3.787 
4.328 
4.869 
5.410 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

58°  W 

57°  45' 

57°  SO' 

57°  15' 

~J 
2 
3 

4 
5 
6 

7 
8 
9 
10 

33°  O' 

33°  15' 

33°  30' 

33°  45' 

.839 
1.677 
2.516 
3.355 
4.193 
5.032 
5.871 
6.709 
7.548 
8.387 

.545 
1.089 
1.634 
2.179 
2.723 
3.268 
3.812 
4.357 
4902 
5.446 

.836 
1.673 
2.509 
3.345 
4.181 
,5.018 
5.854 
6.690 
7.527 
8.363 

.548 
1.097 
1.645 
2.193 
2.741 
3.290 
3.838 
4.386 
4.935 
5.483 

.834 
1.668 
2.502 
3.336 
4.169 
5.003 
5.837 
6.671 
7.505 
8.339 

.552 
1.104 
1.656 
2.208 
2.760 
3.312 
3.864 
4.416 
4.967 
5.519 

.831 
1.663 
2.494 
3.326 
4.157 
4.989 
5.820 
6.652 
7.483 
8.315 

.556 
1.111 
1.667 
2.222 

2.778 
3.333 
3.889 
4.445 
5.000 
5.556 

57°  W 

5«°  45' 

56°  30' 

56°  15' 

34°  0' 

34°  15' 

34°  &W 

34°  45' 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

.829 
1.658 
2.487 
3.316 
4.145 
4.974 
5.803 
6.632 
7.461 
8.290 

.559 
1.118 
1.678 
2.237 
2.796 
3.355 
3.914 
4.474 
5.033 
5.591 

.827 
1.653 
2.480 
3.306 
4.133 
4.960 
5.786 
6.613 
7.439 
8.266 

.563 
1.126 
1.688 
2.251 
2.814 
3.377 
3.940 
4.502 
5.065 
5.628 

.824 
1.648 
2.472 
3.297 
4.121 
4.9.45 
5.769 
6.593 
7.417 
8.241 

.566 
1.133 
1.699 
2.266 
2.832 
3.398 
3.965 
4.531 
5.098 
5.664 

.822 
1.643 
2.465 
3.287 
4.108 
4.930 
5.752 
6.573 
7.395 
8.216 

.570 
1.140 
1.710 
2.280 
2.850 
3.420 
3.990 
4.560 
5.130 
5.700 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

— 

56°  0' 

55°  45' 

55°  30' 

55°  15' 

~T 

2 
3 
4 
5 
6 
7 
8 
9 
10 

35°  0' 

35°  15' 

35°  30' 

35°  45' 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

IT 

.819 
1.638 
2.457 
3.277 
4.096 
4.915 
5.734 
6.553 
7.372 
8.192 

.574 
1.147 
1.721 
2.294 
2.868 
3.441 
4.015 
4.589 
5.162 
5.736 

.817 
1.633 
2.450 
3.267 
4.083 
4.900 
5.716 
6.533 
7.350 
8.166 

.577 
1.154 
1.731 

2.309 
2.886 
3.463 
4.040 
4.617 
5.194 
5.771 

.814 
1.628 
2.442 
3.256 
4.071 
4.885 
5.699 
6.513 
7.327 
8.141 

.581 
1.161 
1.742 

2.323 
2.904 
3.484 
4.065 
4.646 
5.226 
5.807 

.812 
1.623 
2.435 
3.246 
4.058 
4.869 
5.681 
6.493 
7.304 
8.116 

.584 
1.168 
1.753 
2.337 
2.921 
3.505 
4.090 
4.674 
5.258 
5.842 

55° 

Dep, 

0' 

54°  45' 

54°  30' 

54°  15' 

Lat, 

Dep. 

Lat. 

Dep, 

Lat. 

Dep, 

Lat, 

D, 

81 


>-39°45' 


TRAVERSE  TABLES, 


50°15'-54° 


D, 

Lat. 

Dep. 

Lat. 

Dep. 

Lat, 

Dep. 

Lat. 

Dep. 

D. 

36 

O7 

36 

15' 

36 

3O' 

36° 

45' 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

.809 

1.618 
2.427 
3.236 

4.045 
4.854 
5.663 
6.472 

7.281 
8.090 

.588 
1.176 
1.763 
2.351 
2.939 
3.527 
4.115 
4.702 
5.290 
5.878 

.806 
1.613 
2.419 
3.226 
4.032 
4.839 
5.645 
6.452 
7.258 
8.064 

.591 
1.183 
1.774 
2.365 
2.957 
3.548 
4.139 
4.730 
5.322 
5.913 

.804 
1.608 
2.412 
3.215 
4.019 
4.823 
5.627 
6.431 
7.235 
8.039 

.595 
1.190 
1.784 
2.379 
2.974 
3.569 
4.164 
4.759 
5.353 
5.948 

.801 
1.603 
2.404 
3.205 
4.006 
4.808 
5.609 
6.410 
7.211 
8.013 

.598 
1.197 
1.795 
2.393 
2.992 
3.590 
4.188 
4.787 
5.385 
5.983 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

54 

O' 

53° 

45' 

53° 

30' 

53° 

15' 

37 

O' 

37° 

15' 

37° 

30' 

37° 

45' 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

.799 
1.597 
2.396 
3.195 
3.993 
4.792 
5.590 
6.389 
7.188 
7.986 

.602 
1.204 
1.805 
2.407 
3.009 
3.611 
4.213 
4.815 
5.416 
6.018 

.796 
1.592 
2.388 
3,184 
3.980 
4.776 
5.572 
6.368 
7.164 
7.960 

.605 
1.211 
1.816 
2.421 
3.026 
3.632 
4.237 
4.842 
5.448 
6.053 

.793 
1.587 
2.380 
3.173 
3.967 
4.760 
5.553 
6.347 
7.140 
7.934 

.609 
1.218 
1.826 
2.435 
3.044 
3.653 
4.261 
4.870 
5.479 
6.088 

.791 
1.581 
2.372 
3.163 
3.953 
4.744 
5.535 
6.326 
7.116 
7.907 

.612 
1.224 

1.837 
2.449 
3.061 
3.673 
4.286 
4.898 
5.510 
6.122 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

53° 

cy 

52° 

45' 

52° 

30' 

52° 

15' 

38° 

<y 

38° 

15' 

38° 

:$o 

38° 

45' 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

.788 
1.576 
2.364 
3.152 

3.940 
4.728 
5.516 
6.304 
7.092 
7.880 

.616 
1.231 
1.847 
2.463 
3.078 
3.694 
4.310 
4.925 
5.541 
6.157 

.785 
1.571 
2.356 
3.141 
3.927 
4.712 
5.497 
6.283 
7.068 
7.853 

.619 
1.238 
1.857 
2.476 
3.095 
3.715 
4.334 
4.953 
5.572 
6.191 

.783 
1.565 

2.348 
3.130 
3.913 
4.696 
5.478 
6.261 
7.043 
7.826 

.623 
1.245 
1.868 
2.490 
3.113 
3.735 
4.358 
4.980 
5.603 
6.225 

.780 
1.560 
2.340 
3.120 
3.899 
4.679 
5.459 
6.239 
7.019 
7.799 

.626 
1.252 
1.878 
2.504 
3.130 
3.756 
4.381 
5.007 
5.633 
6.259 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

52 

0' 

51° 

45' 

51° 

30' 

51 

15' 

39° 

w 

39 

15' 

39° 

30' 

39° 

45' 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

.777 
1.554 
2.331 
3.109 

3.886 
4.663 
5.440 
6.217 
6.994 
7.771 

.629 
1.259 
1.888 
2.517 
3.147 
3.776 
4.405 
5.035 
5.664 
6.293 

.774 
1.549 
2.323 
3.098 
3.872 
4.646 
5.421 
6.195 
6.970 
7.744 

.633 
1.265 
1.898 
2.531 
3.164 
3.796 
4.429 
5.062 
5.694 
6.327 

.772 
1.543 
2.315 
3.086 
3.858 
4.630 
5.401 
6.173 
6.945 
7.716 

.636 
1.272 
1.908 
2.544 
3.180 
3.816 
4.453 
5.089 
5.725 
6.361 

.769 
1.538 
2.307 
3.075 
3.844 
4613 
5.382 
6.151 
6.920 
7.688 

.639 
1.279 
1.918 
2.558 
3.197 
3.837 
4.476 
5.116 
5.755 
6.394 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

51° 

0' 

50° 

45' 

5O° 

30' 

50° 

15' 

D, 

Dep, 

Lat, 

Dep, 

Lat, 

Dep, 

Lat. 

Dep. 

Lat 

D, 

40°-43°45' 


TRAVERSE  TABLES. 


46°15'-50° 


D, 

Lat, 

Dep. 

Lat, 

Dep. 

Lat, 

Dep. 

Lat. 

Dep. 

D. 

4O° 

<y 

40° 

15' 

40° 

30' 

40^ 

45' 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

.766 
1.532 

2.298 
3.064 
3.830 
4.596 
5.362 
6.128 
6.894 
7.660 

.643 
1.286 

1.928 
2.571 
3.214 

3.857 
4.500 
5.142 

5.785 
6428 

.763 
1.526 
2.290 
3.053 
3.816 
4.579 
5.343 
6.106 
6.869 
7.632 

.646 
1.292 
1.938 
2.584 
3.231 
3.877 
4.523 
5.169 
5.815 
6.461 

.760 
1.521 

2.281 
3.042 
3.802 
4.562 
5.323 
6.083 
6.844 
7.604 

.649 
1.299 
1.948 
2.598 
3.247 
3.897 
4.546 
5.196 
5.845 
6.494 

.758 
1.515 
2.273 
3.030 
3.788 
4.545 
5.303 
6.061 
6.818 
7.576 

.653 
1.306 
1.958 
2.611 
3.264 
3.917 
4.569 
5.222 
5.875 
6.528 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

50° 

O' 

49° 

45' 

49° 

30' 

49° 

15' 

41° 

w 

41° 

15' 

41° 

3O' 

41° 

45' 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

.755 
1.509 
2.264 
3.019 
3.774 
4.528 
5.283 
6038 
6.792 
7.547 

.656 
1.312 

1.968 
2.624 
3.280 
3.936 
4.592 
5.248 
5.905 
6.561 

.752 
1.504 
2.256 
3.007 
3.759 
4.511 
5.263 
6.015 
6.767 
7.518 

.659 
1.319 
1.978 
2.637 
3.297 
3.956 
4.615 
5.275 
5.934 
6.593 

.749 
1.498 
2.247 
2.996 
3.745 
4.494 
5.243 
5.992 
6.741 
7.490 

.663 
1.325 
1.988 
2.650 
3.313 
3.976 
4.638 
5.301 
5.964 
6.626 

.746 
1.492 
2.238 
2.984 
3.730 
4.476 
5.222 
5.968 
6.715 
7.461 

.666 
1.332 
1.998 
2.664 
3.329 
3.995 
4.661 
5.327 
5.993 
6.659 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

49° 

<y 

48° 

45' 

48° 

3O' 

48° 

15' 

42 

O' 

42° 

15' 

42° 

30' 

42° 

45' 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

.743 
1.486 
2.229 
,2.973 
3.716 
4.459 
5.202 
5.945 
6.688 
7.431 

.669 
1.338 
2.007 
2.677 
3.346 
4.015 
4.684 
5.353 
6.022 
6.691 

.740 
1.480 
2.221 
2.961 
3.701 
4.441 
5.182 
5.922 
6.662 
7.402 

.672 
1.345 

2.017 
2.689 
3.362 
4.034 
4.707 
5.379 
6.051 
6.724 

.737 
1.475 
2.212 
2.949 
3.686 
4.424 
5.161 
5.898 
6.636 
7.373 

.676 
1.351 
2.027 
2.702 
3.378 
4.054 
4.729 
5.405 
6.080 
6.756 

.734 
1.469 
2.203 
2.937 
3.672 
4.406 
5.140 
5.875 
6.609 
7.343 

.679 
1.358 
2.036 
2.715 
3.394 
4.073 
4.752 
5.430 
6109 
6.788 

] 
2 
3 
4 
5 
6 
7 
8 
9 
10 

48° 

0' 

47° 

45' 

47° 

30' 

47° 

15' 

43° 

0' 

43° 

15' 

43° 

30' 

43° 

45' 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

.731 
1.463 
2.194 
2.925 
3.657 
4.388 
5.119 
5.851 
6.582 
7.314 

.682 
1.364 
2.046 
2.728 
3.410 
4.092 
4.774 
5.456 
6.138 
6.820 

.728 
1.457 
2.185 
2.913 
3.642 
4.370 
5.099 
5.827 
6.555 
7.284 

.685 
1.370 
2.056 
2.741 
3.426 
4.111 
4.796 
5.481 
6,167 
6.852 

.725 
1.451 
2.176 
2.901 
3.627 
4.352 
5.078 
5.803 
6.528 
7.254 

.688 
1.377 
2.065 
2.753 
3.442 
4.130 
4.818 
5.507 
6.195 
6.884 

.722 
1.445 
2.167 

2.889 
3.612 
4.334 

5.057 

5.779 
6.501 

7.224 

.692 
1.383 
2.075 
2.766 
3.458 
4.149 
4.841 
5.532 
6.224 
6.915 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

47° 

O' 

46° 

45' 

46° 

30' 

46° 

15' 

D, 

Dap. 

Lat, 

Dep. 

Lat. 

Dep, 

Lat. 

Dep. 

Lat. 

D, 

83 


44<>-450 


TRAVERSE  TABLES. 


45°-46° 


D. 

Lat, 

Dep, 

Lat, 

Dep. 

Lat, 

Dep, 

Lat. 

Dep, 

D. 

443  (y 

44a  15' 

44° 

3^ 

44°  45' 

1 

.719 

.695 

.716 

.698 

.713 

.701 

.710 

.704 

1 

2 

1.439 

1.389 

1.433 

1.396 

1.427 

1.402 

1.420 

1.408 

2 

3 

2.158 

2.084 

2149 

2.093 

2.140 

2.103 

2.131 

2.112 

3 

4 

2.877 

2.779 

2.865 

2.791 

2.853 

2.804 

2.841 

2.816 

4 

5 

3.597 

3.473 

3.582 

3.489 

3.566 

3.505 

3.551 

3.520 

5 

6 

4.316 

4.168 

4.298 

4.187 

4.280 

4.205 

4.261 

4.224 

6 

7 

5.035 

4.863 

5.014 

4.885 

4.993 

4.906 

4.971 

4.928 

7 

8 

5.755 

5.557 

5.730 

5.582 

5.706 

5.607 

5.682 

5.632 

8 

9 

6.474 

6.252 

6.447 

6.280 

6.419 

6.308 

6.392 

6.336 

9 

10 

7.193 

6.947 

7.163 

6.978 

7.133 

7.009 

7.102 

7.040 

10 

463  <K 

45    45> 

45°  3W 

45°  15' 

45°  (X 

45°  15' 

45°  3O' 

45°  45' 

1 

.707 

.707 

.704 

.710 

.701 

.713 

.698 

.716 

1 

2 

1.414 

1.414 

1.408 

1.420 

1.402 

1.427 

1.396 

1.433 

2 

3 

2.121 

2.121 

2.112 

2.131 

2.103 

2.140 

2.093 

2.149 

3 

4 

2.828 

2.828 

2.816 

2.841 

2.804 

2.853 

2.791 

2.865 

4 

5 

3.536 

3.536 

3.520 

3.551 

3.505 

3.566 

3.489 

3.582 

5 

6 

4.243 

4.243 

4.224 

4.261 

4.205 

4.280 

4.187 

4.298 

6 

7 

4.950 

4.950 

4.928 

4.971 

4.906 

4.993 

4.885 

5.014 

7 

8 

5.657 

5.657 

5.632 

5.682 

5.607 

5.706 

5.582 

5.730 

8 

9 

6.364 

6.364 

6.336 

6.392 

6.308 

6.419 

6.280 

6.447 

9 

10 

7.071 

7.071 

7.040 

7.102 

7.009 

7.133 

6.978 

7.163 

10 

45^  <y 

44°  45' 

44°  3W 

44°  15' 

D. 

Dep, 

Lat. 

Dep, 

Lat, 

Dep, 

Lat, 

Dep, 

Lat, 

D, 

MISCELLANEOUS    TABLE. 

DIAMETER  =  1.  LOG'. 

Circumference  of  circle,  IT, 3.14159  0.49715 

Area  of  circle, 78540  9.89509-10 

Contents  of  sphere, 52360  9.71900-10 

Earth's  equatorial  radius,  in  miles,     .     .       .3962.57  3.59798 

Earth's  polar  radius,  in  miles, 3949.324  3.59652 

Compression,  1 -<-  299. 1528, 0.00334  7.52411-10 


EQUIVALENTS. 

American,  mile  =         .86756   nautical    miles,     .     .     .  9.93830-10 

"     =1609.40831   meters, 3.20667 

"             "     ==          .21689   German  geosraph.  miles,  9.33624-10 

"             "     =        1.50866   Russian   versts,     .     .     .  0.17859 

yard—         .91444   meters, 9.96115-10 

"     =         .48217   Vienna  klafter,     .     .     .  9.68320-10 

foot=         .30481    meters, 9.48403-10 

"    =         .15639  toises, 9.19421-10 

"    =          .93835    Parisian  feet,    ....  9.97236-10 

"    —         .96435   Vienna  feet,     ....  9.98423-10 

"    =       1.09395  Spanish  feet,    ....  0.03900 

84 


MERIDIONAL   PARTS. 


Deg, 

0' 

10' 

20' 

30' 

40' 

50' 

0 

0.0 

9.9 

19.9 

29.8 

39.7 

49.7 

1 

59.6 

69.5 

79.5 

89.4 

99.3 

109.3 

2 

119.2 

129.2 

139.1 

149.0 

159.0 

168.9 

3 

178.9 

188.8 

198.8 

208.7 

218.7 

228.6 

4 

238.6 

248.6 

258.5 

268.5 

278.4 

288.4 

5 

298.4 

308.4 

318.3 

328.3 

338.3 

348.3 

6 

358.3 

368.3 

378.2 

388.2 

398.2 

408.2 

7 

418.3 

428.3 

438.3 

448.3 

458.3 

468.3 

8 

478.4 

488.4 

498.4 

508.5 

518.5 

528.6 

9 

538.6 

548.7 

458.8 

568.8 

578.9 

589.0 

10 

599.1 

609.2 

619.3 

629.4 

639.5 

649.6 

11 

659.7 

669.8 

680.0 

690.1 

700.2 

710.4 

12 

720.5 

730.7 

740.9 

751.0 

761.2 

771.4 

13 

781.6 

791.8 

802.0 

812.2 

822.5 

832.7 

14 

842.9 

853.2 

863.4 

873.7 

884.0 

894.2 

15 

904.5 

914.8 

925.1 

935.4 

945.7 

956.1 

16 

966.4 

976.7 

987.1 

997.5 

1007.8 

1018.2 

17 

1028.6 

1039.0 

1049.4 

1059.8 

1070.2 

1080.7 

18 

1091.1 

1101.6 

1112.0 

1122.5 

1133.0 

1  143.5 

19 

1154.0 

1164.5 

1175.1 

1185.6 

1196.1 

1206.7 

20 

1217.3 

1227.9 

1238.5 

1249.1 

1259.7 

1270.3 

21 

1281.0 

1291.6 

1302.3 

1313.0 

1323.7 

1334.4 

22 

1345.1 

1355.8 

1366.6 

1377.3 

1388.1 

1398.9 

23 

1409.7 

1420.5 

1431.3 

1442.1 

1453.0 

1463.8 

24 

1474.7 

1485.6 

1496.5 

1507.4 

1518.4 

1529.3 

25 

1540.3 

1551.3 

1562.3 

1573.3 

1584.3 

1595.4 

26 

1606.4 

1617.5 

1628.6 

1639.7 

1650.8 

1661.9 

27 

1673.1 

1684.3 

1695.5 

1706.7 

1717.9 

1729.1 

28 

1740.4 

1751.7 

1762.9 

1774.3 

1785.6 

1796.9 

29 

1808.3 

1819.7 

1831.1 

1842.5 

1854.0 

1865.4 

30 

1876.9 

1888.4 

1899.9 

1911.4 

1923.0 

1934.6 

31 

1946.2 

1957.8 

1969.4 

1981.1 

1992.8 

2004.5 

32 

2016.2 

2028.0 

2039.7 

2051.5 

2063.3 

2075.2 

33 

2087.0 

2098.9 

2110.8 

2122.7 

2134.7 

2146.7 

34 

2158.6 

2170.7 

2182.7 

2194.8 

2206.9 

2219.0 

35 

2231.1 

2243.3 

2255.5 

2267.7 

2279.9 

2292.2 

36 

2304.5 

2316.8 

2329.2 

2341.5 

2353.9 

2366.4 

37 

2378.8 

2391.3 

2403.8 

2416.3 

2428.9 

2441.5 

38 

2454.1 

2466.8 

24795 

2492.2 

2504.9 

2517.7 

39 

2530.5 

2543.3 

2556.2 

2569.1 

2582.0 

2594.9 

40 

2607.9 

2621.0 

2634.0 

2647.1 

2660.2 

2673.3 

41 

2866.5 

2699.7 

2713.0 

2726.3 

2739.6 

2752.9 

42 

2766.3 

2779.8 

2793.2 

2806.7 

2820.3 

2833.8 

85 


MERIDIONAL   PARTS. 


Deg. 

0' 

10' 

20' 

30'      40' 

50' 

43 

2847.4 

2861.1 

2874.8 

2888,5 

2902.2 

2916.0 

44 

2929.9 

2943.7 

2957.6 

2971.6 

2985.6 

2999.6 

45 

3013.7 

3027.8 

3042.0 

3056.2 

3070.4 

3084.7 

46 

3099.0 

3113.4 

3127.8 

3142.3 

3156.8 

3171.3 

47 

3185.9 

3200.5 

3215.2 

3230.0 

3244.7 

3259.6 

48 

3274.5 

3289.4 

3304.3 

3319.4 

3334.4 

3349.6 

49 

3364.7 

3380.0 

3395.2 

3410.6 

3425.9 

3441.4 

50 

3456.9 

3472.4 

3488.0 

3503.7 

3519.4 

3535.1 

51 

3550.9 

3566.8 

3582.8 

3598.7 

3614.8 

3630.9 

52 

3647.1 

3663.2 

3679.6 

3696.0 

3712.4 

3728.9 

53 

3745.4 

3762.0 

3778.7 

3795.4 

3812.2 

3829.1 

54 

3846.0 

3863.1 

3880.1 

3897.3 

3914.5 

3931.8 

55 

3949.1 

3966.6 

3984.1 

4001.7 

4019.3 

4037.0 

56 

4054.8 

4072.7 

4090.7 

4108.7 

4126.9 

4145.1 

57 

4163.3 

4181.7 

4200.2 

4218.7 

4237.3 

4256.0 

58 

4274.8 

4293.7 

4312.7 

4331.7 

4350.9 

4370.1 

59 

4389.4 

4408.9 

4428.4 

4448.0 

4467.7 

4487.5 

60 

4507.5 

4527.5 

4547.6 

4567.8 

4588.1 

4608.6 

61 

4629.1 

4649.8 

4670,5 

4691.4 

4712.4 

4733.5 

62 

4754.7 

4776.0 

4797.5 

4819.0 

4840.7 

4862.5 

63 

.4884.5 

4906.5 

4928.7 

4951.0 

4973.5 

4996.0 

64 

5018.8 

5041.6 

5064.6 

5087.7 

5111.0 

5134.4 

65 

5158.0 

5181.7 

5205.5 

5229.5 

5253.7 

5278.0 

66 

5302.5 

5327.1 

5351.9 

5376.9 

5402.1 

5427.4 

67 

5452.8 

5478.5 

5504,3 

5530.3 

5556.5 

5582.9 

68 

5609.5 

5636.3 

5663.2 

5690.4 

5717.7 

5745.3 

69 

5773.1 

5801.1 

5829.3 

5857.7 

5886.3 

5915.2 

70 

5944.3 

5973.6 

6003.2 

6033.0 

6063.1 

6093.4 

71 

6124.0 

6154.8 

6185.9 

6217.2 

6248.9 

6280.8 

72 

6313.0 

6345.5 

6378.2 

6411.3 

6444.7 

6478.4 

73 

6512.4 

6546.8 

6581.5 

6616,5 

6651.8 

6687.6 

74 

6723.6 

6760.1 

6796.9 

6834.1 

6871.7 

6909.7 

75 

6948.1 

6987.0 

7026.2 

7065.9 

7106.1 

-7146.7 

76 

7187.8 

7229.3 

7271.4 

7313.9 

7357.0 

7400.6 

77 

7444.8 

7489.5 

7534.8 

7580.7 

7627.0 

7674.3 

78 

7722.1 

7770,5 

7819.6 

7869.4 

7919.9 

7971.1 

79 

8023.1 

8075.9 

8129.5 

8184.0 

8239.3 

8295.4 

80 

8352.5 

8410.6 

8469.6 

8529.7 

8590.8 

8653.0 

81 

8716.3 

8780.9 

8846.6 

8913.6 

8981.9 

9051.6 

82 

9122.7 

9195.3 

9269.4 

9345.2 

9422.7 

9501.9 

83 

9583.0 

9666.0 

9751.1 

9838.3 

9927.8 

10019.6 

84 

10114.0 

10211.0 

10310.8 

10413.6 

10519.6 

10628.8 

85 

10741.7 

10858.4 

10979.2 

11104.3 

11234.2 

11369.1 

CORRECTIONS    FOR   MIDDLE    LATITUDE. 


DIFFERENCE  OF  LATITUDE. 

Mid. 
Lat, 

2? 

3 

r 

5° 

6 

7C 

8 

9 

10° 

11" 

12n 

13' 

14° 

15' 

16C 

17° 

18 

19° 

20° 

Mid. 
Lat. 

15° 

, 

2' 

3! 

5' 

7' 

9' 

12' 

15' 

18' 

22' 

26' 

31' 

36' 

41' 

47' 

52' 

59' 

65' 

72' 

15°. 

16 

2 

8 

4 

6 

9 

11 

14 

18 

21 

25 

30 

34 

39 

44 

50 

56 

62 

69 

16 

17 

2 

s 

4 

6 

8 

11 

14 

17 

20 

24 

28 

33 

38 

43 

48 

54 

60 

66 

17 

18 

1 

3 

4 

6 

8 

10 

13 

16 

20 

23 

27 

32 

36 

41 

46 

52 

58 

64 

18 

19 

1 

8 

4 

6 

8 

10 

13 

16 

19 

22 

26 

30 

35 

40 

45 

50 

56 

61 

19 

20 

1 

2 

4 

5 

7 

10 

12 

15 

18 

22 

25 

29 

34 

38 

43 

48 

54 

60 

20 

21 

1 

2 

4 

5 

7 

9 

12 

15 

18 

'21 

25 

29 

33 

37 

42 

47 

52 

58 

21 

22 

1 

2 

4 

5 

7 

9 

12 

14 

17 

21 

24 

28 

32 

36 

4L 

46 

51 

56 

22 

23 

1 

2 

3 

6 

7 

9 

11 

14 

17 

20 

23 

27 

31 

35 

40 

45 

50 

55 

23  , 

24 

1 

2 

3 

5 

?• 

9 

11 

14 

16 

20 

23 

27 

31 

35 

39 

44 

49 

54 

24 

25 

1 

2 

3 

5 

7 

9 

11 

13 

16 

19 

23 

26 

30 

34 

39 

43 

48 

53 

25  -. 

26 

1 

3 

5 

0 

8 

11 

13 

16 

19 

22 

26 

30 

34 

38 

42 

47 

52 

26 

27 

1 

2 

3 

5 

6 

8 

11 

13 

16 

19 

22 

25 

29 

33 

37 

42 

47 

52 

27 

28 

1 

2 

8 

5 

6 

8 

10 

13 

16 

18 

22 

25 

29 

33 

37 

41 

46 

51 

28 

29 

1 

2 

3 

5 

6 

8 

10 

13 

15 

18 

21 

25 

28 

32 

37 

41 

46 

51 

29 

30 

1 

2 

3 

5 

6 

8 

10 

13 

15 

18 

21 

25 

28 

32 

36 

41 

45 

50 

30 

31 

1 

2 

3 

6 

6 

8 

10 

12 

15 

18 

21 

24 

28 

32 

36 

40 

45 

50 

31 

32 

0 

1 

2 

3 

4 

6 

8 

10 

12 

15 

18 

21 

24 

28 

32 

36 

40 

45 

50 

32 

33 

0 

1 

2 

3 

1 

8 

8 

10 

12 

15 

18 

21 

24 

28 

32 

36 

40 

45 

49 

33 

34 

0 

1 

2 

:. 

4 

8 

8 

10 

12 

15 

18 

21 

24 

28 

32 

36 

40 

45 

49 

34 

85 

0 

1 

2 

3 

4 

6 

8 

10 

12 

15 

18 

21 

24 

28 

32 

36 

40 

45 

49 

35 

3o 

1 

2 

3 

4 

6 

8 

10 

12 

15 

18 

21 

24 

28 

32 

36 

40 

45 

49 

36  ' 

37 

1 

2 

3 

4 

6 

8 

10 

12 

15 

18 

21 

24 

28 

32 

36 

40 

45 

49 

37 

38 

1 

2 

3 

4 

I) 

8 

10 

12 

15 

18 

21 

24 

28 

32 

36 

40 

45 

50 

38 

39 

1 

2 

3 

4 

(i 

S 

10 

12 

15 

18 

21 

24 

28 

32 

36 

40 

45 

60 

39 

40 

1 

2 

3 

5 

B 

8 

10 

13 

15 

18 

21 

25 

28 

32 

36 

41 

45 

50 

40 

41 

1 

2 

3 

.) 

8 

8 

10 

13 

15 

18 

21 

25 

•JH 

32 

37 

41 

46 

51 

41 

42 

1 

2 

3 

5 

6 

8 

10 

13 

15 

18 

22 

25 

2!) 

33 

37 

41 

46 

51 

42 

43 

1 

2 

3 

5 

(i 

8 

10 

13 

16 

18 

22 

25 

2!) 

33 

37 

42 

46 

-.52 

43 

44 

1 

2 

3 

5 

ti 

S 

10 

13 

16 

19 

22 

25 

20 

33 

38 

42 

47 

&i 

44 

45 

1 

2 

3 

5 

(i 

8 

11 

13 

16 

19 

22 

26  i  30 

34 

38 

43 

48 

53 

45 

46 

1 

2 

3 

5 

(i 

S 

11 

13  16 

19 

22 

26 

30 

34 

38 

43  48 

••53 

46 

47 

1 

2 

3 

5 

7 

9 

11 

13 

16 

19 

23 

2<i 

30 

35 

39 

44  1  49 

54 

47 

48 

1 

2 

3 

5 

- 

!) 

11 

14 

17 

20 

23  27 

31 

35 

40 

44  I  50 

55 

48 

49 

1 

2 

3 

5 

" 

9 

11 

14 

17 

20 

23 

27 

31 

36 

40 

45 

50 

56 

49 

50 

1 

2 

4 

5 

- 

9 

11 

14 

17 

20 

24 

28 

32 

36 

41  46 

51 

57 

50 

5L 

1 

>) 

1 

5 

1) 

12 

14 

17 

21 

24 

as 

32 

37 

42  47 

52 

58 

51  ' 

52 

1 

2 

4 

5 

- 

9 

12 

15 

18 

21 

25 

29 

3:i 

38 

43  .48 

53 

59 

52 

53 

1 

2 

4 

5 

7 

10 

12 

15 

18 

21 

25 

29 

31 

38 

43  >49 

54 

60 

53  : 

54 

1 

2 

4 

5 

7 

10 

12 

15 

18 

22 

26 

30 

34 

39 

44  |50 

56 

62 

34 

j 

* 

55 

1 

2 

4 

6 

8 

10 

13 

16 

19 

22 

26 

31 

35 

40 

45 

51 

57 

63 

55 

56 

1 

3 

4 

6 

8 

10 

13 

16 

19 

23 

27 

31 

36 

41 

46 

52 

58 

65 

56  ; 

57 

1 

3 

4 

6 

8 

10 

13 

16 

20 

24 

28 

32 

37 

42 

48 

54 

60 

66 

57  i 

58 

2 

3 

4 

6 

8 

11 

14 

17 

20 

24 

28 

33 

38 

43 

49 

55 

61 

<58 

58  ' 

59 

2 

3 

4 

6 

8 

11 

14 

17 

21 

25 

29 

34 

39 

45 

50 

57 

63 

70 

50 

60 

•2 

3 

4 

6 

9 

11 

14 

18 

22 

26 

30 

35 

40 

46 

52 

58 

65 

72 

60 

61 

•1 

3 

5 

7 

0 

12 

15 

18 

22 

26 

31 

36 

42 

47 

53 

60 

67 

75 

61  , 

62 

2 

3 

5 

7 

9 

12 

15 

19 

23 

27 

32 

37 

4:', 

49 

55 

62 

70 

77 

62 

63 

2 

3 

5 

7 

10 

12 

It) 

20 

24 

28 

33 

89 

44 

51 

57 

64 

72 

80 

63 

64 

.) 

3 

5 

7 

10 

13 

10 

20 

24 

29 

34 

40 

46 

52 

59 

67 

75 

83 

64 

65 

2 

a 

5 

7 

10 

13 

17 

21 

25 

30 

36 

41 

48 

54 

62 

69 

78 

86 

65 

66 

2 

3 

5 

8 

11 

14 

18 

22 

26 

32 

37 

50 

57 

64 

72 

81 

90 

66  . 

67 

i 

2 

4 

6 

S 

11 

14 

IS 

23 

28 

33 

39 

45 

52 

59 

07 

76 

85 

94 

67 

68 

i 

2 

4 

6 

8 

12 

15 

1!) 

24 

29 

34 

40 

47 

51 

62 

70 

79 

89 

99 

68 

69 

1  2 

4 

6 

9 

12 

w 

20 

25 

30 

36 

42 

49 

57 

65 

74 

83 

93 

104 

69 

70 

1 

2 

4 

6 

9 

13 

16 

21 

26 

32 

38 

44 

52 

60 

68 

7S 

88 

98 

110 

70 

87 


ECLECTIC  EDUCATIONAL  SERIES. 

Published  by  VAN  ANTWERP,  BRAGG  &  CO.,  Cincinnati  and  New  York. 

THALHEIMER'S  HISTORICAL  SERIES. 

By  M.  E.  THALHEIMER,  Teacher  of  History  and  Composition  in 
Packer  Collegiate  Institute.  For  Graded  Schools,  High  Schools, 
Academies,  and  Colleges.  These  books  furnish  to  Teachers,  stu- 
dents and  general  readers  the  best  brief  and  economical  course  in 
Ancient,  Modern  and  English  History. 

ECLECTIC  HISTORY  OF  THE  UNITED  STATES. 

lamo.,  half  roan,  392  pp.  Copiously  illustrated  with  Maps,  Portraits,  etc. 
Contains  reliable  References  and  Explanatory  Notes ;  Declaration  of  Indepen- 
dence ;  Constitution  and  Questions  on  the  same  ;  Synopses  of  Presidential  Ad- 
ministrations, etc. 

THALHEIMER'S  HISTORY   OF  ENGLAND. 

121110.,  288  pp.  A  compact  volume,  comprehensive  in  scope,  but  sufficiently 
brief  to  be  completed  in  one  school  term.  Its  statements  of  historical  facts  are 
based  upon  the  studies  of  the  most  recent  and  reliable  authorities.  Reliable 
Maps  and  pictorial  illustrations. 

THALHEIMER'S  GENERAL  HISTORY, 

I2mo.,  355  pp.  Maps  and  pictorial  illustrations.  The  wants  of  common 
schools,  and  those  of  higher  grade  unable  to  give  much  time  to  the  study  of 
history,  are  here  exactly  met.  The  teacher  is  aided  by  Revierv  Questions  at 
the  end  of  each  principal  division  of  the  book,  and  by  references  to  other  works 
in  which  each  subject  will  be  found  more  fully  treated. 

THALHEIMER'S  ANCIENT  HISTORY. 

A  Manual  of  Ancient  History  from  the  Earliest  Times  to  the  fall  of  the 
Western  Empire,  A.  D.  476.  8vo.,  full  cloth,  365  pp.,  with  Pronouncing  Vo- 
cabulary and  Index.  Illustrated  with  Engravings,  Maps  and  Charts. 

In  compliance  with  a  demand  for  separate  Histories  of  the  Early  Eastern 
Monarchies,  of  Greece  and  Rome,  an  edition  0/THALHEIMER's  MANUAL  OF 
ANCIENT  HISTORY**  three  Parts  has  been  published,  viz: 

1.  Thalheimer's  History  of  Early  Eastern  Monarchies. 

2.  Thalheimer's  History  of  Greece. 

3.  Thalheimer's  History  of  Rome. 

The  First  embraces  the  pre-classical  Period  and  that  of  Persian  Ascendency. 
The  Second,  Greece  and  the  Macedonian  Empires.  The  Third,  Rome  as 
Kingdom,  Republic  and  Empire. 

Each  part  sufficiently  full  and  comprehensive  for  the  Academic  and  Univer- 
sity Course,  Liberally  illustrated  with  accurate  Maps.  Large  8vo.,  full  cloth, 

For  convenience  the  numbering  of  pages  and  chapters  corresponds  with 
that  of  Thalheimer's  Ancient  History,  so  that  these  separate  volumes  can  be 
used  in  classes  partially  supplied  with  the  complete  work. 

THALHEIMER'S  MEDIAEVAL  AND  MODERN  HISTORY. 

A  Manual  of  Mediaeval  and  Modern  History.  8vo.,  cloth,  uniform  with 
Thalheimer's  Ancient  History.  455  pp.,  and  very  full  Index.  Numerous 
double-page  Maps.  A  sketch  of  fourteen  centuries,  from  the  fall  of  one  empire 
at  Ravenna  to  the  establishment  of  another  at  Berlin. 


36159 


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